How do you tell if a stray is really a stray?

Hi everyone! This is my first post but I hope all you lovely people can help me out!

My question is how can I tell a stray is really a stray or is just passing through the house everyday? Bit of a back story, me and my boyfriend live with his parents, who used to have a cat, who sadly passed away at the age of 19 last year. Left a big hole, but he constantly kept the house and garden his. About 6 months after he passed a grey tabby cat started visiting the house and garden. She was skittish, you saw her only for a few minutes then she would run. One night really late, she was sat in the freezing cold outside the house, we had just come back from work. 

This night was a little different. She still wouldn't let me near her but I HAD to as her left paw looked to be bleeding. In the end I had the lay on my stomach, perfectly still and she sat closer to me. But if I moved a little bit she'd run again. She disappeared for a bit then, but came back and started hanging around the house. she's never been allowed in but we would sit outside and stroke her for a bit until she'd decide she was nervous and run off.

Fast forward to the last month and the temprature has dropped dramtically, rain, freezing cold and we have a cat sat outside, drenched and almost shivering meowing to be let in. Let her in and now she comes around without fail every mornign around 8AM and leaves during the middle of the afternoon. Twice now (including to tonight) she has stayed here all night as well and just fallen asleep, but it is so cold out there tonight! 

She has no collar, looks reasonably healthy, but has a very low hanging stomach (if you get what I mean). None of us thinks shes over 3 years old, and occasionally if you move too quick or if there are people she hasnt met or too many people she runs to a quieter room. 

Just unsure at the moment if she is a stray or just uses the house during the day.

(oh we have a cat flap that is locked from the outside, so she can get out but not in without us knowing!)

Thanks for all your help!

Thank you for helping this cat.

Based on your description, she sounds like she was at one time someone's pet but was lost/abandoned.

Do you want to make her a permanent house cat?  I would keep on doing what you're doing in so far as gaining her trust.  Play soft music when she is inside, try to play with her (feather at end of stick); put in some Feliway.  Let her dictate petting sessions.

A higher priority is to take her to the vets for a check up; make sure the paw has healed, she no doubt has worms and if not already should be spayed.  (I'm assuming the cat is female--some people refer to the low hanging stomach as a spay sway.)

Do let us know how things progress; we love stories like this!

Thanks for your reply, all useful information! Can only reply quick as of to work, will try and get a photo next time I see her. She was round earlier for a short time and then went outside to hunt I think. It was a nice day though.

Thanks for your help and will keep you updated.