If a cat lives completely outdoors can you take it in as your own?

I've lived in my apartment building for almost a year and there has been a cat who is always outside of my building. There is a neighbor who I'm almost positive moved in after me (meaning they could not have brought the cat with them) who leaves cat food and water outside their door. I am almost positive this cat is outside all the time because I work late nights and walk my dog around 3am-5am those days and walk her at normal times on non-work days. It is extremely rare that I do not see this cat on walks. She always seems interested in me and my dog and comes out of no where to watch us from a far. When she sees us returning to the building she returns to this narrow pathway that she knows we always pass through on the way home. She lets me and my dog (she's super cat friendly, I have one already and she used to live with 4) pass super close to her here. And even on the rare occasion my dog lunges at her to play, she never seems phased and seems to want to be close to us.

Today when I went to the dumpster the cat followed me back 2 feet away from me at all times. If I stopped the cat would stop. If I walked, the cat would walk. It followed me up 3 flights of stairs and to my door. And cried outside it when I went inside. I've had 2 landlords who fed an outdoor cat that lived on the premises. Both were really timid for very long periods of time at first but once they started following me, it was only a matter of time before they became extremely loving towards me and started leaving me rodents, etc at my door. This cat seems to really want my attention and comes close to accepting love before she gets nervous and puts 2 ft between us. But then she still sits there contemplating coming to me until I try to walk away and she follows again.

In about a month I'm moving to a new apartment without roommates, it's in the same complex but a decent distance away. I will be living alone which would make introducing a new cat to my pets easier and it would allow me to separate the animals until they are comfortable. Would it be messed up if I take in the cat? The family that feeds the cat don't speak English, I've tried to ask them about the cat in the past. I don't know what to do and if it would be messed up to take this cat in if it seems like it wants to be.

Also when I first moved in, one day while bringing my dog down the stairs I heard a cat crying. When I looked downstairs the cat was staring at me while crying. Once I made it downstairs, the cat ran up to me crying and rubbing itself on me for love. It showed no fear of my 80lb dog and flipped on its back and let me rub its stomach. This cat was so underweight and seemed in need of help. I rushed back upstairs to put my dog away because there wasn't much I could do with her in the way. When I went downstairs the cat was gone and I couldn't find it anywhere. This cat looked exactly like the cat that had been watching me all this time, except this cat is a healthier weight. I am confused if it is the same cat (because they obviously didn't have an owner or had an extremelyneglectful owner) because maybe once this family started feeding it, it got to a healthier weight. And it's a tabby which is common, but the energy seems the same. Because if it is the malnourished cat, then it definitely didn't have a home or it had an abusive home.

What do I do? I feel like this is unnecessarily long, and I'm sorry for that, but I wanted to give the details on why I'm so confused if this cat has an owner or not. I'm also confused if it's wrong to keep a cat outside 24/7 even if it seems to want to come into homes. Help. I want to do the right thing.