Best moves being innacurate (stockfish contradicting itself during analysis)

So I have noticed this for multiple times now, but it has happened so often that my curiosity got the better of me...
Why does Stockfish often contradict itself?

> lichess.org/F7C3jLUf
6. h3?!

1. 6. h3 is in theory the best refutation for stafford gambit
2. 6. h3 is what the browser Stockfish 13+ NNUE suggests using it's cloud analysis
3. After 6. h3 Stockfish even increases the evaluation from +1.8 to +2.0 indicating that the move was potentially even better (because of extra depth after the move was made)

At the same time the game analysis marks the move it just suggested as an innacuracy and continues to give a +1.5 at best for the 2nd best alternative 6. f3

Is there a logical explanation (either technical or a simplified one) for this phenomenon?

Server-side evaluation is set to a limited number of nodes, it's a bit shallow. If you would let the engine analyze deeper it would see that the move is fine. Anyhow, since the the engine returns a lower score than it had on a previous move, the move is marked as dubious, without it necessarily being that.
Client-side analysis runs on your machine and can go deeper and thus has no such problem.