If you have all the pieces and can put it back together like a puzzle I would do this. Except I would use regular wood glue or Gorilla glue. I would use finish nails over wood screws.
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Wood glue, the broken out piece(s) and screws.
If it were me, and I'm a very handy guy, I'd glue/screw it, and I'd finish the shit out of it so you'd never be able to tell. But this would probably be the worst option if you're not picky and can't make it look right. I'd make sure the strike and deadbolt plates were solidly anchored into more than the flimsy door jamb.
If I were recommending to a friend who is not handy at all, the armorconcepts thing doesn't seem hard to install, and it will do the job very well (the latch and deadbolt should be very solid). I don't know what it will look like when it's done though. Maybe it will look fucked up to the landlord.
If I were the landlord, I'd want you to replace the jamb. I wouldn't accept option A, because I wouldn't believe you could make it so I couldn't tell. And I wouldn't want option b because it will look like we have a security problem, or somebody kicked in the door.
When you put it all back together use at least 2" screws for the plates to get into the 2x4 framing or else you'll have the same repair to do again.
Flexseal.
If it was their interior bedroom door, I would offer prevention advice. Remove the door from the hinges and store away. The little bastards hate that shit. Ask me how I know.