[I tried the food inside the door trick.]
I should have told you never close the door and back away with no sudden movement.
It definitely got to their idea to come inside, you could try it again with food that a cat can’t resist but keep putting it a little further inside every time.
But you said the cat allows you to pet her now and if you were able to get her to vet without scaring her, sounds to me she trusts you now.
I have found, at least with Patches, you can tell by their eyes, they will show trust.
When I was trying to get a cat to trust me, I would put the food out on the porch, and then sit on the porch railing and “watch his back” when he ate. He got to where he quite expected it, and even after he came in the house he liked to have some one watch his back while he ate.
We were outside once when a huge raccoon came strolling up the sidewalk aiming for the food, I grabbed the cat, yelled for my husband, and rushed inside all at the same time. He knew we were okay after that incident. Jeez, I loved that cat!