I find the Arthur comparison to Garth Greenhand surprisingly apt. As you point out, GG exists as a number of different characters. With ASOIAF, I’m never sure what to do with Martin’s coy in-world-historical-skepticism, but I can imagine a Maester trying to tease apart the threads of the legend, and if you asked him: “was there a historical Garth Greenhand,” he might answer: “depends what you mean.”

GG is supposedly: the founder of House Gardener, the leader (or a leader) of the First Men, the father of a lot of other heroes, and a fertility god. The first three (and maybe the fourth!) are all things that definitely happened in Westerosi history – someone was the first Gardener, the First Men into Westeros surely had leaders, the great heroes had fathers (whether or not they were all the same man). From this much material you could spin out a thousand different guesses at a “historical” GG … you could even question whether any of them actually had to be named Garth.

Thus with Arthur. If there was anything like Geoffrey’s version – “King Arthur”, ruling much of Britain and fighting Saxons – then his absence from the historical record is astonishing. (But if you were going to lose a guy like that, 5th-6th c. Britain is where you’d do it.) The earlier references to Arthur present one of two themes: a warleader (not necessarily a king) fighting foreign enemies, or a culture hero akin to Finn McCool or Paul Bunyan. One is part of the historical narrative about Romano-British resistance to foreign incursions, and the other rides around the countryside lopping the heads off giants, sometimes being a giant himself, and having bits of landscape named after where his horse stopped for a drink.

Given that even the nature of the historical context in which Arthur-warleader is found is up for debate, “was there a real Arthur” is up for infinite re-definitions, most of which have to land on “maybe” for an answer. I only see two ways to get a “no” – one is to insist that anything short of Geoffrey doesn’t count, the other is to argue that Arthur was a purely fictional culture hero, who was eventually historicized and attached to a bare minimum of historical events but not to any one man’s deeds (because then you could say that he was the “real” Arthur). At the extreme you wind up with cranks doing bad history and worse linguistics telling you that the REAL Arthur was prince of some valley in Wales or Scotland, not named Arthur, and never fought anybody except other princes of valleys in Wales or Scotland, and ohmygod who cares.

But I still want to know: who was the historical Garth Greenhand?

Good question!

I guess I’d say that I see a couple key differences between Arthur and Garth Greenhand. 

1. as far as we can see, there isn’t the same problem of non-contemporaneous sources – the legends of Garth Greenhand are really, really old and the Citadel has preserved runic records that go all the way back to the arrival of the First Men in Westeros, so we’re not relying on, say, post-Andal sources as we might have thought prior to WOIAF.  
2. there’s a relative consistency about Garth. Man or God, pretty much all of the sources say Garth was one of the luminaries of the Age of Heroes, that he had the green clothing, the association to agriculture and fertility, that he was the father of kings and lords and heroes. Indeed, one of the things that I find most interesting that @goodqueenaly brought up is that there’s not even any debate about the birth order of Garth’s kids – no rival ever thought to argue that their ancestor was actually the oldest kid, and everyone seems to agree on who the main kids of the Greenhand were. 
3. there’s an immediacy of the claims of descent. Again, as @goodqueenaly reminds us, it’s not like there weren’t royals who claimed descent from King Arthur, but we don’t really see that happening until almost a millenia after and those claims are pretty clearly modelled after Geoffrey of Monmouth and much later sources. But in House Gardener we have a case where we have heirs of Garth Greenhand from very early on  – judging by regnal numbers, there must have been at least 23 generations of Gardeners before the arrival of the Andals. 

So who was the historical Garth Greenhand? I’m not sure. Could be him:

Or him:

Or him: