When to allow greek gift?

As the white pieces in this position, the lichess computer analysis wants me to step into a greek gift sac by advancing the h pawn. Instead I played a pawn push in the center which is 1.3 eval worse than the push that allows greek gift sac.

What should I learn from this computer analysis? One possible reaction is to dismiss the computer's suggestion as an idea that a human should not necessarily follow, because computers are notoriously better than humans at defending after accepting gambits. A second potential reaction is to re-evaluate how scared I should be of this kind of sac that leaves my king so exposed. Maybe I should not be scared of it on the basis of positional concerns but rather I should only be scared if I can work out tactics that show its danger? A third more subtle response could be to re-evaluate the danger in light of the global board position. Maybe I shouldn't be so scared in this position if the relative lack of development of the black pieces does not justify the sac? Please let me know how you would assess this situation, and how best to learn from it to adjust my intuition!

9. h3 Bxh3 isn't technically a "greek gift" sacrifice. A "greek gift" sacrifice is when the bishop takes a pawn with check, then a knight follows with another check (typically on g5).

If 9. ...Bxh3 10. gxh Qxh, I think 11. Ng5, followed by Qf3 is pretty safe for white.

Yes SLeaver is correct. On a deep analysis of the classic bishop sacrifice on h7(or h2) I recommend looking at Vukovic's "Art of Attack in Chess" which goes into great detail. But Bxh3 here is not a Classic Bishop Sac/"Greek gift".

Thanks for the correction. I had thought greek gift was pretty much any time a bishop sacrifices itself to capture a pawn on the h-file when the opponent is castled kingside.

@brochess I think you played well.
There are no general rules. When your opponent pushes the h-pawn with the intention of pushing it all the way down to h3/h6 you always have to decide anew whether you want to stop the h-pawn advance with h3/h6 or just let your opponent push his pawn to h3/h6 (or even block it with h4/h5).

In your case h3 was possible, because black didn't have enough pieces except for the queen after Bxh3. So your king was safe.
On the other hand playing h3 requires you to calculate on EACH move you make whether the sacrifice is possible.
This will always force you to invest time and energy to calculate the consequences.

The computer evaluation doesn't really matter, because you have to rely on your positional and tactical understanding during a game to find your moves.
You already assessed the position correctly. You wrote "Maybe I shouldn't be so scared in this position if the relative lack of development of the black pieces does not justify the sac?"
You're absolutely right. Steinitz or Lasker said that an attack can only occur from a position of strength and in your game, had no developement, no central control, no developement to back up his attack.
Your weak g2-square could potentially be used for a kingside attack, but that's unlikely because in order to exploit that he has to open the position e.g. the a8-h1 diagonal which is not possible right now because you have a strong grip on the center.

So your assesment and intuition were completely correct. The computer move h3 is not practical from a human point of view, while your move g3 avoids potential sacs and line openings. It's the more practical move. You proceeded with e5 which was correct because you had a lead in developement and therefore you have to open the position.

Nice combination at the end btw :)