Thanks guys, I really appreciate your responses. And what a relief. I was worried that I had inadvertently pulled it out of the differential. I had visions of having to pull the differential and fix it.

The really good news is that I don't think it's from the new seal I installed. I think it's from the old one I removed. The first tool I used to try to pull the seal out was a Lisle seal puller, the kind that looks a little bit like a claw hammer, with metal hooks on both sides. It didn't work, and in the process I bent the metal part of the seal. I think the hook on the seal puller must be what put the crimp in that spring. I ended up borrowing a slide hammer type seal puller from O'Reilly's and it worked great (just like the one Timmy the Tool Man used in his video).

Anyway, I checked the old seal and there is no spring on it, so that must be where the spring came from.