Carrier
The short answer
Whoever moves the waste. Carrying other people's waste as a business needs upper tier registration; lower tier covers a narrower set of cases, including carrying your own.
In practice
A carrier moves waste, it does not store or treat it. The moment a carrier tips into their own yard they are acting as a receiver at that site, and the site needs a permit that allows it. Plenty of operators are both, which is why the two registrations and the permit are separate things.