So I brought, or well called a stray cat I was feeding to my door so my cat sees her; my cat started walking towards the stray cat with her tail up while the cat was staring with it’s tail slowly waving
I closed the door since it seemed suspicious but were they about to fight?
Meeting another cat, especially in their own space, is a stressful thing for them.
You’ll want them to greet each other without being able to get into conflict, such as being able to smell each other under a closed door or maybe one getting to watch the other from a safe distance.
Once they’re not actively starting fights (or getting floofy tails, which is a sign of aggression/fear) and seen interested in each other, then you may be able to have them meet each for limited amount of time in neutral zones.
It takes a long time to get cats integrated with each other.
Obviously if you haven’t done it already, have your stray cat checked out by a vet so you don’t introduce a sickness to your other cat, such as FIV.