“It does seem, however, that bhikkhu and bhauddhayā mean the same thing: one who strives to stop bhava; but bhikkhu is someone who has left home, and bhauddhayā is a householder, correct?”
Yes. That is correct.
Another related word is “sangha”: “san” + “gha” where “gha” also means to remove. Thus “sangha” is someone who has started removing “san”, i.e., one who is Sotapanna Anugami or above (attha purisa puggala).
– Thus a bhikkhu is not necessarily included in “Sangha”. A bhauddhayā could be included in Sangha if he/she is an attha purisa puggala.
– However, these days, “Sangha” is being used just for bhikkhus. That is not quite correct.