A custodian keeps the keys and keeps the books. The holder has a contractual right to the bitcoin; the custodian has the ability to move it. Which supervisor watches the custodian, and what the contract says about theft, insolvency and lending, is the whole question.
Custody can sit under state money-transmitter licences, under a state trust charter, or under a federal charter such as a national trust bank supervised by the Office of the Comptroller of the Currency. Block's application to the OCC describes "consolidating Block's existing bitcoin custody operations under a federally chartered trust bank framework."
When the Gazette reports a custody arrangement it asks four things: who holds the keys, who bears loss, whether the coins can be lent or pledged, and who supervises the answer.