What defines a western route? On almost all of its route the TX Eagle travels west of the Mississippi River, and I always thought that defined western America?
The Texas Eagle has always been considered a western route, but Amtrak didn't bring back traditional dining to it for whatever reason. Your definition of west/east is correct.
It's unofficial, but some of the definition goes back to the Interstate Commerce Commission. The country was divided into regions: Western, Eastern, and Southern. The regions were used for statistical purposes and rate-making, etc. (In earlier years there were five regions.)