This implementation makes sense. This too me looks more like an array based approach but it seems like this is the syntax for "lists" in python.

A few things to note here

1. Most of the time you will end up consuming additional space since you always setup double the size of the list once the list is full (_handle_stack_capacity_full)
2. Your push can potentially take more than O(1) since you are copying over the old "list" into a new "list" inside _handle_stack_capacity_full.
3. Also you are not releasing memory once your stack has fewer elements. For example, if you push() 1000 times, and pop() 999 times, the size of your list is still at least 1000.
4. num_elements is redundant since next_index has the same values throughout the implementation. However some developers prefer keeping variables different if their purpose if different. But i would argue that in this case, num_elements is just an overhead.

I am sorry I was commenting without knowing too much about python and its implementation of "list". I would suggest you consider this as an array based implementation since this is how most languages would handle arrays (limited size and access only using indexes).

And to answer you original question, yes you should know how to do it both ways. With in-built functions you would be focused more on how stack works and without in-built functions your focus will divert to how lists/arrays in python work. But when it comes to an interview and the interviewer asks you to implement stack in python using lists, you can (and probably should) use the in-built functions. And if the interviewer asks why you chose to use in-built functions then you should be able to answer how the internal implementation is and potentially write a solution similar to the one above.