I never owned a cat before but a few months ago I had a similar experience with a cat turning up (or practically living) in our garden. He now lives with us full time and, after plenty searching, found no owners or traces of him having any other human connections. He probably thinks of himself as an outside cat and when he wants to go out we let him out but he stays home a lot just by choice. Especially now the weather is getting cold. I'm no expert but I'd say just let him do what he wants to do and be there for him if he needs you. Don't try too hard to make him be the kind of cat you want him to be. That isn't meant to sound harsh as you obviously have the best of intentions but some cats just like to be outside. Like I say, be there for him with a warm home and a lot of petting if he needs it!
Good on you for making the effort though no matter where he chooses to spend his time!

I'm starting to think the same way. He's just going to be himself no matter what. I'll just have to take him the way he came--on his terms.