The Ring belt and lineal are different, you can trace lots of differences going way back, however, they really went astray after stopping their ratings system in the late 80s. When they restarted they ignored their own history which threw up nonsense like Roy Jones being "THE man" at lightheavy even though the true linage could be traced to Michalczewski (so Roy was Ring champion but not lineal).

Fans constantly confuse the two as they often are in alignment when new linages are created as the top two in a division are almost always the standout best.

As for no.1-vs-no.3 - the Wlad-Chagaev situation happened because of Vitali being rated no.2. However, Bute was Ring no.1/2 (?) when Ward-Froch fought for the title. Pretty sure there are several cases in recent years (can't think off the top of my head though).