A router has multiple IPs. Its public IP is, of course, public. That is the address that you give to any website you connect to so the server knows where to send a reply. A router also has a gateway IP, which is typically the address on your LAN which other devices behind the router see. This IP is private in that it does not exist on the public internet, but it is not secret (most people use the same range).
My question is, is the router's IP really public?
It is public, but you don't have to worry about it. Knowledge of an IP address is typically not a problem unless you need anonymity. You don't have to worry about being hacked or tracked down based on IP. I explained in another answer why there's no need to worry about having a public IP address. If the address wasn't public, then you wouldn't be able to hear a response from any site you connect to!
Or is there another level of NAT in my ISP's network?
Some ISPs have another level of NAT called Carrier-Grade NAT, or CGN. If your public address as seen by the router (not by websites that check your IP) is in the range of 100.64.0.0/10 (see RFC 6598), then you are most definitely using CGN. If you have any doubts, call your ISP and ask them.