Why is 9. Be3 wrong and 9. Nxd4 right?

Thanks in advance for your thoughts on this!

There's both good and bad results from both moves.
nxd4 cxd4; Ne2: The Bg7 is restricted, possible e5 tricks, more qside space/mobility, and (unlikely) f4-f5 with attacking prospects. The black pawn on d4 cramps white a little in the center.
Be3: Threatens to win a pawn. After Bg4 two minor pieces are exchanged (relieves black's cramped position). After Ng4, white might have to surrender the two bishops (and his good to boot).
This is a closed position which I'm not really good at, but these are the important things that I would consider. Just due to my Bg2 looks more active than his Bg7 would make me prefer Nxd4.

as far as i understand, black gains a very weak d pawn after Ne2

Thanks guys for your valuable insights!

In my eyes this decision strategically comes down to the question whether you'd prefer to trade your knight or your DSB for the knight on d4, since apparently you played Be3 with the idea of Bxd4. The knight has to be the better choice, since the DSB is considering the structure c4-d3-e4 your good bishop, which you really don't want to give up, unless there is a good reason and additionally to that the two knights you had after taking with the bishop kind of struggled to find active squares, since Ne2 and Nb5 were pointless in view of e5.

This leads me to another, tactical point. The Bishop on e6 is misplaced, as it prevents black from securing his eventual d4 pawn with e5. By postponing the exchange on d4 by one move you allow the Bishop to relocate itself and make e5 possible. After 9. Nxd4 cxd4 10. Nb5 defending d4 would have been a lot more awkward for black.

@gartae In order to improve faster. First and most important is to avoid moves like 39. Rbh2. Hanging pieces is an important issue and demands much more attention and effort than the move you asked about. If you don't make any obvious mistakes you'll improve much more than by knowing strategical subtleties. There is a difference but not at the level your game was played. By now you could safely play both moves. In general it is considered that B>N so it's better to keep the bishops on the board. They will become active, eventually, and are better than knights.

Nxd4 is better because the knight is a powerhouse on d4. It's going to need to be exchanged at some point. That's a must. No c or e pawn can boot it away ever from that outpost. There is a good chance that the d pawn can be won after this being up a pawn and still having a bishop pair.

Be3 on the other hand... You are going to run into Bg4. You may end up losing that bishop pair as well by snapping the knight off with your bishop. The d4 pawn is no longer as soft. Although with this move you develop a tiny bit faster than the other lines getting that bishop in play sooner. I would not say it's a bad move, but there is better. It's playable.

So Nxd4 is the right move.

On the other hand something is to be said of the move h3 from here. It keeps the tension and adds some looft while being prophylactic in stopping a knight or bishop from going to g4. This is also a good move from here.

I would say that all 3 moves are playable. And in terms of strength they go: Nxd4 > h3 > Be3

Thanks to all of you!