Sets from top to bottom:
1. Countably infinite/Countable (included here for completeness)
2. omega finite
3. Stackel finite
4. Tarski finite
5. Bounded, Streamless or Noetherian (From intuitionist set theories)
6. Amorphous
7. Δ2
8. Δ3, which includes Motowoski finite
9. Δ4
10. Δ5
11. Kuratowski finite
12. Hyperset (from non well founded set theories)
13. Fuzzy set
(NB I knew they exists, but I am too lazy to find the axioms that knocks out the Deltas one by one)
I wonder if there is a way to prove, that omega finite is the smallest possible finite in the hierarchy of finite definitions...