You’re absolutely right. I’m on my phone trying to imagine the scenarios in my head and trying to find the simplest possible example.

Maybe 4 participants is too few to clearly see what happens.

(A,X),(B,Y) as the proposals is fine for the point being made. The issue is if we reverse which group is the proposer, we still end up with the same matching, because (A,X) strictly prefer each other.

The minimal example with 4 participants is if each person P has unique preferences and P's 1st choice has P as 2nd choice. So in this case just flip X:

A:XY B:YX X:BA Y:AB

If A,B propose you get AX,BY. If X,Y propose you get BX,AY. Both are stable because the proposers are getting their first choice.