After castling, capture with f or h pawn?

Consider the following position:
lichess.org/editor?fen=5rk1/6pp/6p1/8/8/6P1/6PP/5RK1_w_Qq_-_0_1
Often times after castling kingside I will recapture a piece on g3 for white/g6 for black away from the center. I will always capture with the f-pawn.
My logic is that I get the open f-pawn. I also am scared of a queen/rook battery along the h-file that could checkmate me.
I do realize that many people recapture w/ the h-pawn.
Am I doing something wrong, if so, can someone explain? Could someone explain why capturing with the h-pawn is better and/or I shouldn't be afraid of the non-existent battery?

I feel like this is something I gotta understand.
thanks :D

Generally people say "bring your pawns to the center. Based on that rule, taking with the h-pawn is generally considered better.

Think about Pawn Structure, and Piece Activity
the way you capture can sometimes create structural weaknesses or create outside passed pawns in the endgame.
Trickier is piece activity; do any of your pieces benefit from the way you capture(usually a rook on f1/f8), do any of your opponents pieces gain anything(because threats can be made on your king).
You probably already thought of that but really that's all there is to it

Your K and P endgames will be all lost. ;)

Your K and P endgames will be all lost. ;)

There are some special opening lines in the French, Slav, QID where fxg3 (...fxg6) is somewhat better. Side-effect: e3 and e6 become terribly weak. Ng4 or Ng5 coming.

There are three aspects, sorted by importance:

1) pawn majorities
2) dispersion of the pawn structure
3) piece play on the half open file.

Some examples:

lichess.org/editor/8/ppp3pp/8/3p4/3Pp3/6b1/PPP2PPP/8_w_-_-
hxg3 is better - prevents a passer on e-file

lichess.org/editor/8/ppp2ppp/8/8/4P3/6b1/PP3PPP/8_w_-_-
hxg3 is better - fxg3 would disperse the pawn structure and destroy the king side majority.

lichess.org/editor/8/ppp2ppp/8/3p4/3P4/6b1/PPP2PPP/8_w_-_-
both moves are good as no pawn majorities are possible and no dispersion of the structure will happen. It depends on how the pieces are placed. If eg. white 0-0ed then fxg3 may be better, otherwise hxg3 to open the h-file for the rook.

lichess.org/editor/8/ppp2p2/8/3p2pp/3P4/5Pb1/PPP2P1P/8_w_-_-
fxg3 is better - prevents a passer on h-file

lichess.org/editor/8/ppp2p2/5p2/3p2p1/3P4/6b1/PPP2PPP/8_w_-_-
fxg3 is better - pressure on backward pawns on f-file, potential passer on h.

See also 'Pawn Power in Chess' by Kmoch, Chapter II, §3

Usually you recapture towards the center, because of the endgame.
That is not written in stone. Together with O-O-O there are often recaptures dxc3 or dxc6 or cxb3 or cxb6 as this increases king safety and helps develop Bc1/Bc8.
E.g. 1 e4 e5 2 Nf3 Nc6 3 Bb5 a6 4 Bxc6 dxc6
1 e4 e5 2 Nf3 Nf6 3 Nxe5 d6 4 Nf3 Nxe4 5 Nc3 Nxc3 6 dxc3
Both recaptures
1 e4 c6 2 d4 d5 3 Nc3 dxe4 4 Nxe4 Nf6 5 Nxf6 exf6 and ...gxf6 are equally viable.
So as a rule you recapture towards the center, but there are many exceptions.