Concerning game name changes:

I don't think of myself as a cooperator or defector; I consider myself to be one who (edit: tries to) cooperates if my opponent cooperates and defects if my opponent defects.

So, if the name changes, that can change what I expect others to do, and thus what I find that I should do.

It's not clear how much of the effect is itself that - the expectation that others will see it differently. I wonder what would happen if you told the two players different things as to what the name of the game was, and then told one or the other player - or both - what the other player had been told the game would be.