You misunderstood me. When the original Sonic (lets call him Solaris Sonic) went back to blow out the flame, he created a new timeline. His original timeline still retains the events of 06 and we stop following Solaris Sonic as there are no more games that take place after 06. In the new timeline, we meet Sonic (lets call him Modern Sonic) in Sonic 1 and he goes through the Classic and Adventure Sagas, but because Solaris doesn’t exist in this timeline anymore, 06 can’t happen and the Rush games take its place. This is the same Modern Sonic that was able to go on to later games like Unleashed, Colors, and Lost World, hence his nickname.
And in case you bring up the whole “Classic Sonic is a separate character” thing, that’s true and false at the same time. Modern Sonic still went through the Classic Saga much like Solaris Sonic did, but the Time Eater picked up Sonic at some point between 3&K and Adventure and forced him to join Modern Sonic for Generations. At the end, he goes back to his timeline with Classic Tails (and Classic Eggman later on) and Mania starts up, creating the Classic Timeline.