The (now retired) shuttle orbiter) is a spaceship as was the Command Modules and Lunar Modules that went to the moon. Any vehicle designed to operate in space is a spaceship. But they are certainly not starships given they are designed to operate within limited parameters.Not all spaceships are starships, but all starships are spaceships.
But a genuine starship, while also by nature isa soaceship, is a specific thing. A starship is specifically designed for interstellar travel beyond a solar system and to other star systems. It needn't be a FTL vehicle to be a starship. It need only be designed to be able to cross interstellar distances. As such even a sleeper ship can be a starship because it's designed fir interstellar travel even though it will take years, decades or centuries to make the trip. A generational worldship, like the Yonada asteroid, is also a starship.
By that reasoning even a vehicle like the shuttlecraft Galileo is a starship because it can cross interstellar distances from one star to another. A TNG/DS9 era runabout is basically a large shuttlecraft.
Today we use different words to describe essentially similar devices: desktop, laptop, tablet and smartphone are all variations of computer. In SF shuttlecraft, sleeper ship, world ship, explorer, etc. are all classifications for different kinds of starships.