Does Tom Bombadil give names to the ponies, or does he just know their true names?

Question

In the "Fog on the Barrow Downs" chapter of The Lord of the Rings, Tom Bombadil retrieves Merry's five ponies, which had scattered whilst the hobbits were in the barrow. He calls them each by name.

Merry...had not, in fact, given them any such names, but they answered to the new names Tom had given them for the rest of their lives.

The narrator uses the word "given" there, but the narration is from the hobbits' perspective, so it may be that Bombadil simply appeared to have given them those names. Did he "assign" names to the ponies, or did he simply know which names they preferred?

Answer 1

Tolkien doesn't specify

For what it's worth, Christopher Tolkien seems to believe the names were chosen by Bombadil, based on this note from The Return of the Shadow (emphasis mine):

Tom Bombadil's names for the ponies go back to the beginning, with the exception of 'Sharp-ears', who was first called 'Four-foot'!

^{History of Middle-earth VI The Return of the Shadow Part One: "The First Phase" Chapter VII: "The Barrow-wight"}

But this is the only note on this subject that I can find; later parts of Return of the Shadow don't mention this at all, and I've found nothing whatsoever in Letters.