Permissioned or Permissionless blockchain? Which one is PoS more adapted for?
Frankly, I don't think your definitions are useful. But by your definitions, PoS makes a network permissioned by definition. In a proof-of-stake network, you can only participate in the validation of transactions if you possess the token, which you can only get from someone else who has the token and chooses to give it you. By your definitions, that makes the system permissioned.

Again, I don't think your definitions are useful.

To propose more helpful definitions:

A system is permissionless or public if it permits open participation in both the submission of transactions and the choice of transactions to make forward progress on. There must be no special permissions required to submit and execute transactions beyond the possession of some way to pay transaction fees, and this must (in actual practice) be widely and easily available to anyone who makes a reasonable effort to acquire it. The method in which the ledger makes forward progress must justify a reasonable expectation that hard censorship will not occur -- that is, anyone who submits validly signed transactions to the network should be able to reasonably expect them to execute without having to worry that some particular clique or group can decide to disallow their transactions specifically.

A system is permissioned or private if it does not permit open participation in either the submission of transactions or the choice of transactions to make forward progress on. That is, submitting a transaction requires some permission beyond mere possession of a widely-available, anti-spam asset or participants cannot reasonably expect the system to resist censorship, that is, not all participants have reasonable assurance that their transactions won't be discriminated against in a way that significantly affects their ability to use the system and get its benefits.