I now see it clearly. You're right. You can't use the same work to extend another chain, unless the new chain is dependent on the old chain, regardless of the configuration of the new chain. The new chain founders can either choose to completely disconnect it with the old chain, or make it based on it.

Forking the chain in the former fashion (e.g., Bitcoin Cash) in Proof-of-Stake does grant the validators the same benefits oppositely to Proof-of-Work, indeed. That being said, you could say that Proof-of-Work makes splitting / forking less desirable.