We don't know whether it was an arbalest or a crossbow.
To quote (for the lack of a better source) Wikipedia:
A large weapon, the arbalest had a steel prod ("bow").
Taking that as the trait that distinguishes an arbalest from a crossbow, we... gain nothing. There is no mention of arbalests in the original Russian version, and descriptions given to crossbows don't indicate the material of the prod.
Ложа Анкиного арбалета была выточена из черной пластмассы, а тетива была из хромистой стали и натягивалась одним движением бесшумно скользящего рычага. Антон новшеств не признавал: у него было доброе боевое устройство в стиле маршала Тоца, короля Пица Первого, окованное черной медью, с колесиком, на которое наматывался шнур из воловьих жил.
Funnily enough, in the translation by Olena Bormashenko1 Anton's crossbow is described as an arbalest, presumably due to copper covering:
The black stock of Anka’s crossbow was made of plastic, while the strings were chrome steel, operated by a single motion of a noiselessly sliding lever. Anton didn’t trust newfangled technology; he had an old-fashioned arbalest in the style of Marshal Totz (King Pitz the First), overlaid with black copper, with a cable of ox sinew wound around a little wheel.
However, only Anton's crossbow is ever described as an arbalest in this translation. Being described as "old-fashioned", I don't think it was really an arbalest, given that in our world arbalests were developed in 12th century, and Marshal Totz, presumably, lived before that.
One could argue that since the world of Hard to Be a God appears to have plate armour, the need for arbalests is obvious. Brother Aba's crossbow, for instance, is described differently:
Брат Аба с неожиданной для его комплекции резвостью извлек из-под стола огромный боевой арбалет и положил перед собой прямо на бумаги.
Brother Aba, with surprising agility for his bulk, took a huge combat crossbow out from underneath the desk and placed it on the papers in front of him.
Chapter 7. Emphasis mine.
Maybe it is an arbalest due to being a combat crossbow, but we don't have any definitive evidence; and it's not clear why it's not described as an arbalest in the translation, while Anton's clearly is.
Therefore, given the ambiguity in authors' descriptions, both are equally valid; "crossbow", however, is more recognisable.
1: I sent her an e-mail asking about her choice of words; she responded that there was no underlying motivation to use "arbalest"
instead of "crossbow".