The names that it uses is the one attached to the Microsoft account. if you change it (or attempt to change it) it'll tell you that this 'new name' will be reflected across your accounts.
I wanted my name to just be 'pm' (since that's what the church calls me) but I didn't want that to be reflected across all my accounts.
Of course, I'd love my gamertag to be pm, but someone who has never used XBOX Live is sitting on it
.
EDIT: Also - your gamertag still shows up - it just shows your real name off to the side. so my name looks like...
techpastorm
Matthew Titus
On my profile.