What is the name of the data structure that is a tree on the backend but has a list like API?
In "Optimal  Algorithms for List Indexing and Subset Rank" by Paul F. Dietz this data structure is called an "indexed list" and an algorithm is given that has complexity $O(\log n / \log\log n)$ for all operations. Unfortunately this term has little value as a search term.

While that is complex data structure note that you can transform any self-balanced tree into the data structure you want. Just replace the key field of each node with a field that keeps track of the number of elements to the left of it, and update this on inserts/deletes (takes $O(\log n)$ time). Then use that field as your key when searching for element $i$ in the tree.

Also check out "tiered vectors".