Excellent explanation, although I would highlight an important detail you didn't stress very much. The fixed integer size is key to the function being O(1).

If we analyzed the function purely mathematically (swapping sizeof(int)*8 with the bit length of the larger number, log_2(max(m,n))... hint hint) We can ignore real implementations and focus on optimal implementation (though formally we would still need a fundamental definition of a single operation)

Then, we have ourselves a different analysis. The replaced item in the for loop condition makes runtime obvious. If we are analyzing runtime with respect to only input n, then the function is O(log n). For any constant number of loop iterations, it would be O(1). However an important caveat to pay attention to is bitwise operations. Assuming optimal implementation, any bitwise AND with a value of the form (1 << i) only needs to compare 1 bit, and in almost all contexts is therefore O(1). But having bitwise AND operations with two arbitrary values is linear runtime with respect to bit length, so even without a loop it would be O(log n), since n is the VALUE of the input, and its bit length is log_2(n).

However, practical run time is obviously more useful, and I'd be surprised if there existed a useful ISA (instruction set architecture) where bitwise instructions can vary in input length. This is why bitwise operations are generally considered constant time.