Just because one of his goals is to drive competition into the race to mars doesn't mean it's his only goal. I don't doubt he not only wants to speed up the race to mars but wants to be known as the first.
That's pretty awesome.
Is Richard Branson not a human?
A suborbital manned spacecraft, OTOH, is a spaceship in every sense. (It's neither an orbital spaceship nor, say, an interplanetary spaceship, but it's definitely a spaceship.)
Saying they aren't is like saying a riverboat isn't a boat because it's not ocean-going.
It can touch space, but a ship is not defined by its ability to touch the water.