If it makes you feel better about it, sure. But after the first month without contact, it was pretty clear what was going to happen.
It gives me a legal leg to stand on if she ever does show up again. And with people like this, you cannot predict what they might think they are entitled to if it becomes convenient to them again.
The pessimist in me makes me point out that with people like this, legal standing may not be that important. But I hope they just stay the ever loving hell away and let you get on with continuing to improve this innocent beast’s life.
Someone upthread mentioned checking for a chip. If you haven’t done that yet, this might be a good time to do it. If the dog doesn’t have one, get it done.
Regarding the legality of it, I don’t know how that would actually play out. I have a feeling that if it came to it, a judge would give the dog back to them. I don’t know if there’s a way to do this without them suddenly deciding they want the dog back, but it would probably help your case if you could send her a text message getting her to confirm the dog is now yours.
Having said that, my gut still says it’s probably best not to contact her at all and just go on with the assumption that the dog is yours. Worry about what to do if she you contacts you when or if that happens.
If you’re going to call it a stray dog, make sure you check on stray/found dog laws in your area. At least where I live, you can’t just keep a dog you found without going through a series of steps, including notifying the owner, if known.
You can get a chip scanner online very inexpensively, so you don’t risk finding out the hard way at the vet when the vet notifies the previous owner. They’re only $20 - $35 on Amazon.