A vomitorium was not, as commonly believed, a room where rich Romans went to induce vomiting in order to eat more, but rather a high-capacity entry/exit into/from a theatre. But how did the notion that vomitoria were for vomiting in come about? Was it purely a matter of mistaken English etymologies?

I can't give a full answer to the main part of the question, but can clarify that the myth goes a layer deeper than you think! The interpretation of 'vomitorium' as a 'vomiting-room' is modern, but it's even worse than you think: 'vomitorium' as a word with a serious architectural meaning is also modern. It dates to the 16th century.

Where ancient architectural writers talk about passages into or out of a theatre, in Latin they're called itinera ('passages') or exitus ('exits'), as in Vitruvius.

There is only one passage in any ancient writer that uses vomitorium to refer to a passage -- in an amphitheatre, not a theatre -- and it's clearly a metaphor based on a poetic allusion, not a technical term. The passage is from the 5th century writer Macrobius, Saturnalia 6.4.3:

‘Each morning the whole building vomits a wave of clients’ (Vergil, Georgics 2.462): ‘vomits a wave’ is a fine expression, and an old one, for Ennius says: ‘and Tiber’s river vomits into the salted sea’ (Ennius, Annals fr. 142 Vahlens). This is why even now we refer to vomitoria at the games, since people enter in a mass and pour into their seats.

~~Vergil's line about a building 'vomiting' people~~ correction: Ennius' line about a river vomiting is in turn an allusion to the Greek epic poet Apollonius of Rhodes, who refers to a river 'vomiting' into the sea (Argonautica 2.744).

That snippet of Macrobius is the only ancient passage that uses vomitorium to describe part of a building, but it wasn't a super-rare word. The adjective vomitorius is a straightforward one with the obvious meaning 'pertaining to vomiting'. It's not hard to see why someone might have thought it referred to a place for vomiting, because if you are crazy enough to use it as an architectural term, then that is the most obvious interpretation.

Now, having said that, that doesn’t mean we must stop using the word when talking about modern theatres. Its history doesn’t dictate its present-day usage. It's just that it wasn’t a standard term for the Romans, so far as we know.

The earliest use of the word as a serious architectural term that I can find is in Philandrier's notes on Vitruvius, though the note was omitted from the initial 1544 publication (here it is in the notes to a 17th century edition of Vitruvius: see footnote c). In English 'vomitorium' as a serious term starts popping up in the 1700s.

I can't tell you exactly when, where, and why it was reinterpreted as a 'vomiting room', but it must have been after that ponit. As I said it's not hard to see why: it's analogous to a number of other words for rooms -- cubiculum 'bedroom', vestibulum 'clothing room', or with closer analogies in bath houses like frigidarium 'cold room/pool', and so on. It may also have been motivated by a passage like this, in Cicero's In defence of King Deiotarus 21:

The prosecutor) goes on, ‘When you (Caesar) said you wanted to vomit after dinner, they started to take you to the bathroom: because that’s where the ambush was. But your perpetual good luck saved you, because you said you’d rather be in your bedroom.’

Caesar's vomiting was for medical reasons, as Cicero tells us in another letter, and obviously in this passage he does it in the bathroom, not a special vomiting room. I did a piece on this a few years ago that has more details -- but I'm afraid it doesn't cover the specific question of how the modern myth of 'vomiting rooms' developed.

Not necessarily, but also yes.

I'm not a latinist so I wouldn't be able to fully digress on the way the word "vomitorium" became associated with that myth, but from the architectural standpoint, "vomitoria" were corridors used (mostly) as what we nowadays call "emergency exits". The Romans were very technical engineers and amphitheatres (like the Colosseum, for instance) were very advanced buildings designed for entertainment. In the larger ones, specially in bigger cities, you could see full re-enactments of past battles and they would be realistic portrayals of it with "pyrotechnics" or even water, if the battle had occurred in the ocean. These bigger ones would also seat a lot of people and in the event of a mishap where people had to escorted out as fast as possible, these corridors would be used.

The word "vomitorium" is almost to a sense a metaphor for the process of getting people out of the structure through a small corridor. In this sense, it was as if people would be "vomited" from the building.

So, it does have to do with vomiting, but not in the sense that it became associated with—which I am pretty sure comes from a misconception and it doesn't only apply to the English language.

I know this won't fully answer your question, but it may help out a little bit, haha.