It’s not been explicitly confirmed, but given the similarity in forms to later notes signed Santa (and confirmed to be Dumbledore), the note’s writer claiming the Cloak was freely given by Harry’s father, and the explicit warning against letting Dumbledore see the Cloak, it’s very likely that Dumbledore gave Harry the Cloak as in canon.
It’s not been explicitly confirmed, but given the similarity in forms to later notes signed Santa (and confirmed to be Dumbledore), the note’s writer claiming the Cloak was freely given by Harry’s father, and the explicit warning against letting Dumbledore see the Cloak, it’s very likely that Dumbledore gave Harry the Cloak as in canon.