You should also take in consideration that from e8 the knight may in some day go to g7 and e6 or to c7 and to e6, so again, I think 9...Ne8 is probably better than Nd7 in this position (I agree with you that in other lines Nd7 is thematic and the main move, for example in the Taimanov lines where the knight does good on covering c5.)
At less than master level it hardly matters which knight move you make because the inevitable mistakes that follow will outweigh any difference in their strength.
But if you really want to figure out which is stronger you cannot decide between Ne8 and Nd7 by general reasoning like this. You have to look at actual lines of play. That is doubly true of an opening like the King's Indian which is very tactical and combative. And not just one or two master games from four decades ago. The King's Indian has moved on a lot since Fischer's time and many lines that he played are now thought to be "not the best".
Of course Fischer could play the same "inferior" lines against 99.9999% (a well-considered percentage) of chess.com's members and still win easily. Heck, if he were still alive he could start with 1. e4 and 2. a4 and he'd still beat most of us.
Not really, in the Bayonet Attack Nh5 is the stronger than Nd7.
In the Taimanov lines Nh5 is imposible, and Nd7 is better than Ne8 because it covers c5 from there, making it harder for White to play c5.
In this case Nh5 is probably better but it is a matter of taste between Nh5 and Ne8 while Nd7 in my opinion is inferior as I have stated in other posts.