It means exactly what it says on the label. The first player to take action, be it reserving a name or creating a character with that name once character creating is activated, gets the name. You don't own any name, ever. - you are using names, so stop calling them "my names," it's silly. The fact the system picked isn't favorable to you doesn't makes it unfair. Fairness is about treating everyone equally, and that's what will be done here.
I don't have an official answer to your question, but my assumption is the server won't have character creation for a limited period of time while transfers are done. People will have a chance to keep more names than just the one(s) reserved, but that will depend on no one else from the other server already having a character with the same name and transferring faster.