I have only made one osage bow, so take it for what it is worth. The thin ring is the spring ring. On the bow I made, this ring was very brittle and could be scrapped off very easily.
You want the backing of the bow(front side of bow) to be the thicker ring(annual ring). You should take it down to the spring ring just before you annual ring, then remove the spring ring by simply scraping it off. This will leave you with a good backing of the annual ring.
As you scrape off wood, you can tell when you hit the spring ring. It is not as fiberous as the annual ring, almost like a condensed powder. The sound of the spring ring coming off is much diferent than the sound of the annual ring coming off.
I am sure someone with more experience will be able to explain this better than I can.
This link may help out some. I was shown this webpage when I was making my bow and it helped alot.
http://residents.bowhunting.net/sticknstring/default.html