I don't recall any other references to owls wanting payment aside from the case from The Philosopher's Stone when Hagrid's owl wanted payment for delivering his message to Dumbledore after making contact with Harry out on the rock island.
Was this payment specifically for the owl? (Thus implying owls use wizard money).
Or was Hagrid's payment ostensibly going to the owl's "keeper" as a sort of rental fee?
Hagrid explains why the owl wanted payment - it was for delivering the newspaper. The newspaper costs money, so the owl delivering it has to collect the money and take it back for the Daily Prophet. It doesn’t seem like the payments are actually for the owls, but that they collect the money for the Daily Prophet and deliver the money to them similarly to how they deliver messages.
“The owl swooped in and dropped the newspaper on top of Hagrid, who didn’t wake up. The owl then fluttered on to the floor and began to attack Hagrid’s coat.
‘Don’t do that.’
Harry tried to wave the owl out of the way, but it snapped its beak fiercely at him and carried on savaging the coat.
‘Hagrid!’ said Harry loudly. ‘There’s an owl –’
‘Pay him,’ Hagrid grunted into the sofa.
‘What?’
‘He wants payin’ fer deliverin’ the paper. Look in the pockets.”
- Harry Potter and the Philosopher's Stone, Chapter 5 (Diagon Alley)
Owls take messages for free, but it seems like there’s a subscription service where wizards can have the Daily Prophet delivered to them by owl and pay when the owl arrives with their paper. Hermione also gets the Daily Prophet and has to pay the owl that brought it.
“Hermione, however, had to move her orange juice aside quickly to make way for a large damp barn owl bearing a sodden Daily Prophet in its beak.
‘What are you still getting that for?’ said Harry irritably, thinking of Seamus as Hermione placed a Knut in the leather pouch on the owl’s leg and it took off again.”
- Harry Potter and the Order of the Phoenix, Chapter 12 (Professor Umbridge)
Another time Hermione is delivered the Daily Prophet, she has to give payment to the owl as well.
“Hermione was not the only person eagerly awaiting her Daily Prophet: nearly everyone was eager for more news about the escaped Death Eaters, who, despite many reported sightings, had still not been caught. She gave the delivery owl a Knut and unfolded the newspaper eagerly while Harry helped himself to orange juice; as he had only received one note during the entire year, he was sure, when the first owl landed with a thud in front of him, that it had made a mistake.”
- Harry Potter and the Order of the Phoenix, Chapter 26 (Seen and Unforeseen)
The Daily Prophet owls presumably then return to the Daily Prophet with the money they’re given to take. The money seems to be payment for buying the newspaper itself, not for the owl delivering it.
It is clearly a payment for the service used. In your example here, the service is mail delivery.
thus implying Owl's use wizard money
No, they don't. At least, there isn't any example or occurrence in the book to support that.
Or was Hagrid's payment ostensibly going to the owl's "keeper" as a sort of rental fee?
Not a rental fee, but a charge for the mail delivery service.
