The theatrical venues are not the performing artists’ company. They’re the names of the venue. I myself am not an English guru... but this conversation is brought up sooooo many times and nobody has a clear answer.
If it helps my point at all, I have an bachelor degree in Theatre Tech. But basically we had a whole class conversation about this one day for improv/debate purposes... and it basically boiled down to “it’s interchangeable. One just sounds more fancy than the other.” And “nobody knows.”
So it is really either or. Whether it is the art form itself, or the physical building. You’ll see so many people using both versions of the word for all categories.