Talk to a locksmith
They have no trouble configuring lock mechanisms to engage in any of those behaviors.
As an example, there used to be a "Classroom lock" that once unlocked by the key cannot be relocked except with the key, that was to prevent smart aleck students from locking teacher out. They are now rolling that back because of school shooters.
Anyway they'll have no trouble getting what you want.
The kind of locksmith you want has a retail location where you can take your locksets to them. Many locksmiths are outbound ONLY and they will cost an arm and a leg. "locksmiths" on the internet are typically hot ral, lead generating sites who sell your name to outbound locksmiths.