No offense intended, but in bullet-ID terminology, those aren't rings, they are grooves. (A "ring" on a bullet's body is a raised ridge or band.)
In my opinion, the lower one appears to be a "crimping" groove, where the lip of a metallic cartridge gripped the bullet. So, despite the unusual cavity in its base, I believe you've got a metallic-cartridge bullet.
In the photos, your Caliper shows the bullet's diamter as being .657 -- an extremely unusual size. It doesn't match up with any known caliber of bullet. But your other photos seem to show some
irregular-shaped "swelling" above the bullet's base. Please re-measure your bullet's diameter just slightly above the swelling.
Then, post your photos & info in the Bullets category at the following link. The Civil War Projectiles forum at bulletandshell.com is the best place on the internet for identification of civil war era (and "slightly" post-civil-war) bullets.
Civil War Projectiles Forum - Index