Space Shuttle isn't a "rocket" like Falcon 9 is though, it couldnt go to space by itself. So saying its the first reusable rocket is really stretching credibility.

Falson 9 is a one piece rocket, as is Superheavy.

The Space Shuttle got to space with the help of other rockets, tank etc.

And all those other parts were recovered after most flights and re-used after refurbishment on future flights. Since there's no other name than "Space Shuttle" for the whole rocket, it will do. Note that Starhsip is also ambiguous, as it refers both to the entire two stage rocket, but also just stage 2.