Yes, I understand that you see it that way and only that way.
And this purse-snatching guy is another great example! You've got a million of 'em.

There are so many possible ways of seeing the incident, in addition to "he lied."
It reminds me of a time when I was in college, and a classmate announced that the pen I was using was hers. When I told her that, no, this was my pen, the one I'd been using all semester, she went to the administration, who scheduled a powwow to confront me. She was very upset, and described how she had borrowed her husband's very special and expensive pen and he would be horribly angry at her if he found out it had been stolen. I calmly explained that it was a promotional give-away from Sears, but that I was glad to give it to her if it would help. I felt pretty humiliated, though, when everyone saw that as me confessing to the crime.
The next day she came to school, handed me the pen, and told me that her husband had looked at the pen, told her that this cheapy thing was not his pen, THIS was his pen -- he had found his designer expensive pen the night before on the entry table. She apologized to me, I accepted, and we became friends.
Imagine if her "You're lying!" had reactivated me into saying "No, YOU're lying!" etc. It certainly looked to her like it was absolutely, 100% objectively, inarguably true that I had stolen her husband's pen. We could have a much more difficult and uncomfortable road to friendship!
And who is to say it was "my" pen, anyway? Who is to say it's "my" purse that gets snatched? The guy may totally believe it's "his" purse for all I know, and for all I know, he may be right! I can't know what's going on in his head. Even if he felt he was lying -- like he consciously knew that that purse belonged to me and yet he said it was his anyway -- there are STILL other perspectives from which to see it as a non-lie.
I understand your resistance to a subjective reality perspective, and that's fine. And in fact, if someone takes something that I "own," it might be helpful to use a subjective lens -- in the real world. Of course, like you, I can stick to my objective reality guns and say: Book him, Dano! That's perfectly valid. But it's no more The Truth than to see it as his purse as well as mine -- his money as well as mine. Or maybe it would be more helpful to notice that if someone takes something away from me, maybe it belongs with him -- maybe he is desperate to feed himself or his kids, and maybe it was time for me to get a new purse, and maybe etc. etc.
Have you ever heard the story of the monk who arrived home to find a burglar stealing everything the monk had? The monk told the burglar --- "go ahead, take what you need." And when the burglar left with all of the monk's stuff, the burglar looked out the window and up at the moon, and thought to himself, "Poor man, I wish I could give him the moon, too."