> What does this word mean? "Villa"? As in a house?
"a house, usually in the countryside or near the sea, especially in southern Europe, and often one that people can rent for a holiday."(From Cambridge Dictionary)
I'd remind you that the「別墅」is what you called amyamy.

> So Xiomicha and Amyamy are just the same as here, but for different groups?
Even though it seems like they're different groups but in fact they're overlap each other to some extent.

> The pattern seems to be that they're named after whomever owns them, yes?
Yes! Correct!

Have a nice trip. 
Wanderer and  Observer.