Every pawn move and every capture resets the count. A quiet move by another piece adds one halfmove, even if it gives check. The fifth field of a FEN records this halfmove clock; it is not the same as the fullmove number in the sixth field.
A correct claim can concern a completed threshold or a declared intended move that will complete it, using the event’s procedure. The player must have the move when making the claim. A supplied FEN counter is an assumption about history, not proof that a claim was made correctly.
FIDE’s 75-move rule is automatic after seventy-five moves by each player without a pawn move or capture. Checkmate on the final move takes precedence. Online claim handling can differ, so check your platform’s rules.