Do specify the zone, maybe everyone has its own rule. but the point is (to logger):
>What is the procedure to transfer a domain to another Registrar? I must ask the current Registrar to have changed >admin-c handle on my nick and after that registrars understand each other?
registrars to understand each other WILL not. you have tech Registrar's request so-called auth code (transfer code) then the new transfer domain, where enter this code, but then they have as you say "agree among themselves", although it all happens automatically
PS: and still call the area, maybe in the TLD — one rules (much more transparent, simpler and more correct), and in local areas their own, where everything is much more complicated