I'am updating/ organizing all my snippets and commands, because I want to
use git in order to "sync" my bundles between my Macs.

Digging into the Package Content, it seems that the Bundle Editor does not
update the [name].tmSnippet or [name].tmCommand when changing it's name! It
keeps the initial name (so sometimes untitled).

This is intentional for ease of version control integration. There is nothing wrong with changing the name yourself though, the only thing that matters from TextMate’s perspective is the UUID in the bundle item.