The Tor Check web page will tell you if you're using Tor or not.
If you're using Tor, your browsing is anonymous and can't be traced back to you. Your ISP may well know you're using Tor - because they can tell that you're connecting to a Tor entry guard - but they won't know what you're using it to look at.
The problem you're probably facing is that NBC are actively detecting Tor traffic, and giving you the error page you're seeing on the assumption that you must be outside the US if you're using Tor. (Which wouldn't necessarily be true, but they can assume whatever they like.) The BBC are doing exactly the same thing with both Tor and VPN traffic. They can do this because the IP addresses of Tor's exit nodes are known. (As are the exit addresses of lots of VPNs.)
Disclaimer: Also note that there are probably legal reasons as to why their content isn't available outside of the US. For example, NBC probably don't own the rights to the content themselves, and therefore can't allow it to be seen outside of the US without the owner's permission.