TL;DR: The assignment of lines to speakers comes down to the judgment of the editor or translator, based on their sense of what is dramatically most appropriate in a given case, and whether their goal is to attempt to reproduce the original performance or not.
In this answer I’m going to investigate what we know about the chorus in ancient Greek tragedy, and how we know it (or think we know it). The evidence is in many cases pretty thin, resting on archaeology, on the surviving texts of plays, on the etymology of words, and on passing mentions in Aristotle and other writers. By the end you should be in a position to evaluate Scully’s choices yourself!
Origin of the tragic chorus
Tragedy originated at the Dionysia, a festival in honour of the sufferings of Dionysus, the god of the grape harvest. This was originally a rural festival, held around the winter solstice, but later had an urban counterpart in Athens, held around the spring equinox. Such festivals probably go back to the seventh century BCE or earlier, for Herodotus says that in the time of Cleisthenes’s grandfather (confusingly also called Cleisthenes), the inhabitants of the nearby town of Sikyon had replaced Dionysus with their local hero Adrastus:
the Sikyonians then not only gave other honours to Adrastos, but also with reference to his sufferings they specially honoured him with tragic choruses, not paying the honour to Dionysos but to Adrastos. Cleisthenes however gave back the choruses to Dionysos.
Herodotus (c. 430 BCE). Histories V.67. Tranlated by G. C. Macaulay. Project Gutenberg.
The Dionysian performances were called τραγῳδία (tragodia), conjectured to derive from τράγος (tragos, he-goat) + ᾠδή (ode, song), perhaps because a goat, an animal commonly sacrified to Dionysus, was the prize. It seems likely that originally these performances consisted of a chorus only, and that actors were introduced gradually, perhaps originating as chorus-leaders. Aristotle says:
The number of actors was first increased to two by Aeschylus, who curtailed the business of the Chorus, and made the dialogue, or spoken portion, take the leading part in the play. A third actor and scenery were due to Sophocles.
Aristotle (c. 335 BCE). Poetics IV. Translated by Ingram Bywater (1920). Oxford: Clarendon.
Though we have to bear in mind that Aristotle was writing a hundred years and more after these events, and may have been reporting tradition or folklore rather than documentary evidence.
Theatre of Dionysus
At some time in the sixth century BCE the Theatre of Dionysus began to be constructed on the Acropolis hill in Athens. It now looks like this, with the lower tiers of seating, the orchestra (the semi-circular platform), and part of the proskenion (the raised part of the stage) remaining.
Photo by dronepicr (2015). Licensed CC-BY-SA.
The ancient form of the theatre is often reconstructed as shown below, where the features are (1) orchestra (2) parodos (3) skene (4) proskenion. But it must have been modified and developed over its period of use, so earlier plays were performed against a more rudimentary structure than pictured here.
Ernst R. Fiechter (1914). Die Baugeschichtliche Entwicklung des Antiken Theaters, p. 191. München: Oskar Beck.
Reconstructions like this are based on archeological evidence at the site, on other ancient theatres in better states of preservation, on contemporary paintings (on walls and pottery), and on the texts of the plays themselves. For example, many tragedies have a scene where palace doors are flung open, or a tent flap is raised, to reveal something dramatic like a murder or suicide (for example, at Aias line 346 where Tekmessa opens the tent to reveal Aias and the cattle he has slaughtered), and so it is thought that the stage had an opening at the back to represent this, called σκηνή (skene) meaning “tent” (Liddell & Scott). Similarly, many plays have a scene in which a god appears suddenly and speaks to the characters (for example, near the end of Philoktetes where Herakles tells Philoktetes that he will be healed if he goes to Troy), and so in Fiechter’s reconstruction there are doors leading out onto the roofs of the parodoi where the actor playing the god might have stood.
How do we know there was a chorus-leader? First, there are mentions in surviving texts, for example:
Then again there is Aristeides of the tribe of Oeneis, who has had a similar misfortune. He is now an old man and perhaps less useful in a chorus, but he was once chorus-leader for his tribe. You know, of course, that if the leader is withdrawn, the rest of the chorus is done for. But in spite of the keen rivalry of many of the chorus-masters, not one of them looked at the possible advantage or ventured to remove him or prevent him from performing.
Demosthenes (348 BCE). ‘Against Midias’, 60. Translated by A. T. Murray (1939). London: Heinemann.
Here the words are κορυφαῖος (koryphaios) meaning “leader of the chorus” (L&S) and χορηγός (choregos) meaning “one who defrays the costs for producing a chorus” (L&S).
Second, there are passages in which an actor in dialogue with the chorus uses a vocative singular, for example, Philoktetes, line 1172: τί μ᾽ ὤλεσας; τί μ᾽ εἴργασαι (Why have you ruined me? What have you done to me?) where τί is vocative singular. These cases suggest that the actor is in dialogue with a single member of the chorus:
In all such cases the actor has focused his attention on the words of the chorus-leader. These vocatives occur in expressions where the actor is referring to the words which the chorus-leader has just spoken, and the contents of the leader’s words arouse a strong emotional reaction in the actor. Thus, the actor’s vocative to the chorus-leader reflects a sudden rise in the intensity of the dialogue.
Maarit Vuorenjuuri (1969). ‘Vocative singular addressing the chorus in Greek drama’. Arctos–Acta Philologica Fennica VI, p. 148.
Third, there are passages in which the chorus addresses itself, indicating that the dialogue has to be split among the chorus members.
What we know is consistent with all the chorus’s spoken dialogue being originally delivered by individual members, but except in a few cases like the ones discussed above we have no evidence either way.
What did the chorus do?
χορός (choros) originally meant “dance” (L&S) and later “band of dancers and singers”. They performed in the ὀρχήστρα (orchestra), which is related to ὄρχησις (orchesis) meaning “dancing” (L&S) and ὀρχηστής (orchestes) meaning “dancer” (L&S). So originally the function of the chorus was to dance. The very size of the orchestra in the Theatre of Dionysus shows that something more spectacular took place than actors standing still and declaiming. We are pretty sure that the chorus sang as well as danced, for example, Aristotle says:
a tragedy has the following parts: Prologue, Episode, Exode, and a choral portion, distinguished into Parode and Stasimon; these two are common to all tragedies, whereas songs from the stage and Commoe are only found in some. The Prologue is all that precedes the Parode of the chorus; an Episode all that comes in between two whole choral songs; the Exode all that follows after the last choral song. In the choral portion the Parode is the whole first statement of the chorus; a Stasimon, a song of the chorus without anapaests or trochees; a Commas, a lamentation sung by chorus and actor in concert.
Aristotle (c. 335 BCE). Poetics XII. Translated by Ingram Bywater (1920). Oxford: Clarendon.
It’s worth looking at the etymologies of the words that Aristotle glosses in this extract. πάροδος (parodos) means “narrow entrance, gangway” (L&S), suggesting that this was sung while the chorus entered the theatre via such an entrance. ἐπεισόδιος (episodios) means “addition” (L&S), suggesting the earliest tragedies did not have these passages of dialogue between the songs. στάσιμος (stasimos) means “standing” (L&S), usually interpreted as referring to songs that were sung when the chorus was already standing on stage. κομμός (kommos) means “beating, especially of the head and breast in lamentation” (L&S).
Scholars generally guess that the chorus sang in unison, and not in harmony or polyphony. The size of the theatre, the outdoor acoustics, and the wearing of masks by the performers means that special effort must have been undertaken to ensure that the words could be made out by the people in the highest tiers of seating.
Identifying the songs
How do we distinguish songs from spoken dialogue in the texts of plays, given that we don’t have any stage directions, and that assignments of speeches to speakers are conjectural? We use the metre. Aristotle says that playwrights represented spoken dialogue using iambic verse:
Tragedy acquired also its magnitude. Discarding short stories and a ludicrous diction, through its passing out of its satyric stage, it assumed, though only at a late point in its progress, a tone of dignity; and its metre changed then from trochaic to iambic. The reason for their original use of the trochaic tetrameter was that their poetry was satyric and more connected with dancing than it now is. As soon, however, as a spoken part came in, nature herself found the appropriate metre. The iambic, we know, is the most speakable of metres, as is shown by the fact that we very often fall into it in conversation, whereas we rarely talk hexameters, and only when we depart from the speaking tone of voice.
Aristotle. Poetics IV. Translated by Ingram Bywater (1920). Oxford: Clarendon.
And indeed we find that classical tragedies alternate between passages that are dialogue-like in iambs, and passages in more poetic diction, in more complex metres suitable for song and dance.
In the passage from Aias asked about in the question, Athena’s closing speech to Odysseus is in iambs, ending at line 133, and then at line 134 the metre switches to anapaests for the chorus (“Son of Telamon” in Scully). An anapaest goes short–short–long, so that it can be sung to music in quadruple time, suitable for the chorus to sing as they march out onto the orchestra. Then at line 172 (“That mother of a rumor” in Scully), the metre changes again to the more complex “dactylo-epitritic”, consisting of dactyls (long–short–short) mixed with second epitrites (long–short–long–long). So starting at line 172 is the lyric ode, sung while dancing. Together these songs form the parodos. Then at line 201, where Tekmessa enters (“Shipmates of Aias” in Scully), the rhythm changes back to iambs, indicating that we are now in the first episode. See Jebb, pp. lx–lxxiii, for metrical analysis of all the songs in the play.
So now you know some theories about how Aias might originally have been performed, where these theories come from, and how tenuous some of the evidence is. It’s possible now to evaluate Scully’s choice to assign some of the lines of the “Son of Telamon” passage at lines 134–171 to the chorus leader, and to write “(severally)” for the remaining lines, indicating that the lines should be divided among the chorus members. The conventional opinion, represented by Richard C. Jebb, is that since this is a marching-song for the entrance of the chorus “it can scarcely be doubted that the anapaests were chanted by the whole Chorus, and not by the coryphaeus alone”, but I am sure that Scully was aware of this, and so his choice to deviate indicates that he is not concerned with attempting to reconstruct the original performance, but had some other purpose. The introduction to the book says “As translators we have a responsibility not to reissue a replica of classical Greek culture but rather to recoup its living reality” and praises the Oxford New Translations as “more natural, idiomatic, and thus more accessible and playable versions”. So my guess is that Scully is concerned with writing a translation that could be performed today, with modern stage conventions, where the cast do not normally sing their lines in unison.