... On the whole, it is better to hit greens in regulation than to miss them. The green was missed, and if we're not counting it as such, then we're missing the opportunity to accurately identify the real opportunity. In this case, the opportunity would lie in the approach shots, while the stats will be showing it to be in putting.
I'll also argue that green size is not completely arbitrary. The green, which is specifically defined by the rules, was specifically designed in that shape and size by the architect for a reason. Nothing arbitrary at all.......
The size and shape of the green is certainly not defined by the rules. It is defined by the architect *originally*, but the sizes and shapes change over the years for many reasons. Greenskeepers or superintendents might decide to shape it differently for various reasons (a tree gets too big say, and provides too much shade in one area which allows fungus to grow - or, they want to make the greens harder for a tournament by narrowing them, or expanding them to make them closer to a hazard). And the guys who ride the mower don't go over the exact same line day after day, year after year - that would be impossible.
More to the point though, they're arbitrary in the sense that no green is exactly the same size and shape as another. If you hit 3 small greens, is your iron play better than if you hit 3 large greens from the same distance? Yes, but either way your stats show 3 greens hit, so how much did the size of the green really matter? If you hit 5 greens with second shots that were the result of really good drives so you were really close, is your iron play better than if you hit the same 5 greens from shots further out? No, but either way your stats show 5 greens hit, so again, how much did the size of the green really matter?
You can also think about it this way: Two golfers have the exact same skills. Golfer A plays all his rounds at courses that have generally large greens. Golfer B plays all his rounds at courses with tiny greens. Whose stats are going to be more accurate? Neither, of course. Golfer A might have better GIRs and higher total putts, but it doesn't matter because he's using the stats to track whether he's getting better or worse in any given area.
When someone counts the fringe as the green, what they're doing is equivalent to playing a different course that's the same in every respect except the greens are a little bigger. Yes, that hypothetical second course might be rated a little differently, but ratings and slope don't enter into stat keeping. You don't adjust your fairways hit for example on a course that has really tight fairways.You'll simply have fewer fairways hit than otherwise, but that doesn't make your stats wrong.