How do I determine the basic operation of this algorithm?
"Basic operations" are whatever you choose. You may get different results based on your choice, which is great as it leads to understanding algorithms better! You may have seen the number of comparisons and swaps being analyzed independently in array sorting algorithms. 

As a corollary, your professor can pick whatever they want. We can't possibly know their mind; you'll have to ask them. 

That said, if you're after "time complexity", you want to count how often a dominant operation occurs. See here and here (section "A note on asymptotic cost"). In case of doubt, just count all candidate operations. 


  Getting different results based on my choice is something I also think is not applicable here, as the two options I am struggling between have identical results in when going towards infinite input sizes.


In that case, the choice doesn't matter. Either both operations are dominant and your result is correct, or they are both not and you need to pick another operation entirely.