For me, not my full name or the common nickname given for my full name. It would depend on my nickname, like how it was spelled. My name is Kimmie, and I can't stand the spelling as Kimmy. I'd accept Kimi, and Kimie. I may accept Kimmie and Kimmi, though I'd be hestitant.
My online names? Hell yes! I would definitely buy the pet Light, or the pet with my neo name, and I already have the pet that uses my other online name I use for mmorpg style games.