For the moment, I suppose you can still make a case for casual use of the term "spaceship" to include suborbital flights. I doubt that will continue for very long as private orbital spacecraft and launch vehicles become less novel; the sub-orbital minor leagues probably won't be seen by most as being the "real thing".

Just ask Alan Shepard. You know, the first American in space?

Presumably because in this sense, the Mercury wasn't "his spaceship" as he didn't own it or the organization that produced it (like Gagarin and the Vostok, for that matter).

Mercury was capable of staying in space. The rocket that put it on a suborbital flight is a different thing.