I think the right word is
reservation.
You reserve a "first name" which have to comply with some set of rules and you choose a common "last name" (extension) which have to comply with another set of rules. The combination makes it unique and thereby reserved for you.
Some names, like trademarks and governmental names are already reserved, even if they are not registered or in use.
In the beginning, domains were mostly free for reservation, then administration was monetized. The fee doesn't mean it's leased, but just the cost of registration and administration. Plus some profit
