I think we should distinguish between "interpretations of probability" and "interpretations of statistics" (or more adequately- "statistical argumentation"). Both, are naturally tightly related.
Regarding interpretations of probability- I like the following criteria
- Single-case or replicable?
- Mental or physical?
- Objective or subjective?
Any interpretation out there can be classified according to these aspects.
Regarding statistical arguments, the classics are indeed the frequentist and the Bayesian but there are more out there-- fiducial, fuzzy logic. Indeed, there are infinitely many ways to argument with data, but the two you mentioned are preferred in the sense they are considered more "convincing".
[a] [Williamson, Jon. "Philosophies of probability." Handbook of the Philosophy of Mathematics 4 (2009).]