What abbeys have you been to? All of the ones I’ve been to don’t let
anyone into the cloister, besides the monks/nuns and those on formal discernment events, and for very good reason; the cloister is to be the monastic’s place away from the world where they can go to rest in the Peace of Christ. That is their vocation, to tend to the Opus Dei; everything else a monastery does is purely incidental, and is not necessary to their life.
Also, I’ve never heard of a monastery, except maybe the Carthusians, but they don’t allow any visitors, barring just women from their monastery; doesn’t mean it doesn’t happen, I’ve just never seen or heard it.
To answer the OP: it depends on the Order. Most monastic Orders, and I believe the Dominicans too, use the process you described, and others don’t allow their new members to take different names.
Here’s a thread that goes into good detail about this topic:
forums.catholic-questions.org/showthread.php?t=545499&highlight=religious+names