I have some general advice when tackling questions about ancient literature, which is to always check primary texts. Secondary sources are vital for understanding and interpreting primary texts, but they often mix up claims for which there is evidence, with claims that are only conjectural or traditional, or which are specific to some circumstance, leaving you uncertain how far you can trust or generalize them.
So in this answer, I’ll look at both of the sources you found and follow references to try to understand what evidence they are based on, and then I’ll demonstrate a corpus-based approach to the question.
If you follow the references, you’ll find that the text you quoted from Wikipedia was copied from Chambers’s Cyclopaedia (1728):
CORYPHAEUS, in the ancient Tragedy, was the Chief or Leader of the Company that compos’d the Chorus. See Chorus.
The Copyphaeus spoke for all the rest, whenever the Chorus took part in the Action, in quality of a Person of the Drama, during the Course of the Acts.
Anon (1728). Chambers’s Cyclopaedia, volume I, p. 334. London: James & John Knapton.
This does not itself have any references, so we must classify it as a rumour, meaning that this is what (some) scholars in the early 18th century believed about the role of the chorus-leader. Also, even if it were correct, it would not help answer your question, because you are asking about choral songs and the Cyclopaedia is discussing choral dialogue.
Here’s a bit more of the context for the claim you quoted from Weiss:
in Aristophanes’ Thesmophoriazusae, for example, the tragic poet Agathon performs his own antiphonal choral ode, in the role of both chorus and chorus leader (101–29)
Naomi A. Weiss (2020). ‘Ancient Greek Choreia’. In Eleonora Rocconi & Tosca A. C. Lynch, eds. A Companion to Ancient Greek and Roman Music, p. 164. Wiley.
You want to know whether Weiss can tell us whether the chorus leader normally sang along with the chorus. That is, to what extent can her claim be generalized? In order to figure this out, we must examine the evidence for Weiss’ claim. The only evidence cited is lines 101–129 of Aristophanes’ Thesmophoriazusae, so let’s look at it in translation, with a bit of the preceding context:
Euripides. Here comes Agathon.
Mnesilochus. Where, where?
Euripides. That’s the man they are bringing out yonder on the machine.
Mnesilochus. I am blind then! I see no man here, I only see Cyrene.
Euripides. Be still! He is getting ready to sing.
Mnesilochus. What subtle trill, I wonder, is he going to warble to us?
Agathon. Damsels, with the sacred torch in hand, unite your dance to shouts of joy in honour of the nether goddesses; celebrate the freedom of your country.
Chorus. To what divinity is your homage addressed? I wish to mingle mine with it.
Agathon. Oh! Muse! glorify Phoebus with his golden bow, who erected the walls of the city of the Simois.
Chorus. To thee, oh Phoebus, I dedicate my most beauteous songs; to thee, the sacred victor in the poetical contests.
Agathon. And praise Artemis too, the maiden huntress, who wanders on the mountains and through the woods …
Chorus. I, in my turn, celebrate the everlasting happiness of the chaste Artemis, the mighty daughter of Latona!
Agathon. … and Latona and the tones of the Asiatic lyre, which wed so well with the dances of the Phrygian Graces.
Chorus. I do honour to the divine Latona and to the lyre, the mother of songs of male and noble strains. The eyes of the goddess sparkle while listening to our enthusiastic chants. Honour to the powerful Phoebus! Hail! thou blessed son of Latona!
Aristophanes. Thesmophoriazusae, lines 97–129. Translated by Eugene O’Neill (1931). Aristophanes Comedies, pp. 273–275. New York: Rarity Press.
Agathon was a tragic playwright who was contemporary with Aristophanes, but none of his works survive. Plato tells us in the Symposium that Agathon “won the prize” at the City Dionysia “with his first tragedy, on the day after that on which he and his chorus offered the sacrifice of victory”.
Remember that assignments of lines to speakers in ancient plays are conjectural, so we should check that the assignments in this passage can be reasonably deduced from the text. It seems clear that Agathon is the singer, because he is named by Euripides, and because both Euripides and Mnesilochus tell us that he is going to sing. The start of the ode is indicated by a change of subject, by a change of register from ordinary speech to elevated poetic language, and by a change of metre (in the Greek original). The ode is in the form of a dialogue between two speakers (this is what Weiss means by “antiphonal”) and so it is reasonable to deduce that the two speakers are Agathon and a chorus of “damsels”.
But why does Weiss tell us that Agathon performs both the roles of chorus and chorus-leader here? Well, in the Thesmophoriazusae, Aristophanes mocks Agathon’s poetic style, pretensions and cross-dressing, and so one way to stage the scene for maximum comic effect would be to have Agathon playing the chorus of damsels as well as the solo part. There is a tradition to this effect, for example O’Neill (1938) adds the stage direction:
Agathon now sings a selection from one of his tragedies, taking first the part of the leader of the chorus and then that of the whole chorus.
However, this tradition is purely conjectural and we can imagine other ways to stage the scene, for example maybe Euripides and Mnesilochus take the chorus parts, or maybe the chorus come on pushing Agathon in the “machine” (probably an εκκύκλημα) mentioned by Euripides.
So, returning to the original question, is Weiss’ claim generalizable? The answer must be “no”—Agathon’s ode in the Thesmophoriazusae is a comic skit and (assuming that it was in fact staged with Agathon singing both parts) it would not be funny if it were typical for one singer to take both parts in an antiphonal ode. We can tentatively deduce that the opposite must have typically been the case: that the two parts in antiphonal odes were taken by two different sections of the chorus. But note that this tells us nothing about the majority of choral odes, which were not antiphonal.
I used the search tool at the Perseus Digital Library to find all relevant mentions of the word κορυφαῖος in their corpus.
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.
Demosthenes (348 BCE). Against Midias 60. Translated by A. T. Murray (1939). London: Heinemann.
In respect of order; such are all things which are systematically arranged in relation to some one determinate object. E.g., he who is next to the leader of the chorus is prior to him who is next but one
Aristotle (4th century BCE). Metaphysics 1018β. Translated by Hugh Tredennick (1933). London: Heinemann.
And indeed actors think the audience do not understand unless they put in something of their own, and so they strike all sorts of attitudes, as you see bad flute-players whirling about if they have to do “the Discus,” or mauling the leader of the chorus when they are playing the “Scylla.”
Aristotle (4th century BCE). Poetics 1461β. Translated by W. H. Fyfe (1932). London: Heinemann.
it necessarily follows that the goodness of all the citizens is not one and the same, just as among dancers the skill of a head dancer is not the same as that of a subordinate leader.
Aristotle (4th century BCE). Politics 1277α. Translated by H. Rackham (1944). London: William Heinemann.
But when many of [the Celts] sup together, they all sit in a circle; and the bravest sits in the middle, like the coryphæus of a chorus; because he is superior to the rest either in his military skill, or in birth, or in riches
Athenaeus of Naucratis (3rd century). Deipnosophistae, IV.36. Translated by C. D. Yonge (1854). London: Henry G. Bohn.
These sources are clear that the chorus-leader held his position by virtue of his skill: he was the best at dancing (Aristotle, Politics) and superior to the rest of the chorus in general (Aristotle, Metaphysics; Athenaeus). It would make no sense for the most skilful member of the chorus to remain silent or stationary when the rest were singing or dancing, and indeed Demosthenes says that “the rest of the chorus is done for” if the leader does not lead them. So I think we can be confident that the chorus-leader sang and danced with the rest of the chorus, except in special cases such as antiphonal odes.