Sure, but a start-up is more of a determination after the fact than something you decide to build. The only reason it works out for YC is because they start so many companies that a few of them due to the expected distribution end up being home runs. They still influence things as much as they can in that direction but there are absolutely no guarantees and quite a few of the go-big-or-go-home companies end up going neither big nor home, they end up being normal companies.
So unless you're willing to apply the start-up label only after the fact to the successful companies you're going to have to be a bit more inclusive than to just use it to describe Facebook, dropbox, google, Uber and AirBnB.
> But startups are qualitatively different - they are usually started with the express (and usually only) purpose of growing very large, very fast.
No, that's the exception. The rule is that start-ups were started to be companies like any other. At some point during their life span they found a groove that supported the property of very fast growth (almost always these are companies with some kind of network effect) and then retrospectively applying your criteria you can call those companies start-ups.
Anyway, no need to believe me, you can simply prove me wrong by deliberately starting a start-up that will scale. I'll bet you 1:50 that you won't make it.
The hard part then is this: your company has a fairly good chance (10% or so) to become just another company. Now supposing this is the case, will you then follow through on your 'go big or go home' slogan and go home and kill that company? Or will you do what everybody else does in that situation and just run it and milk the cow?
And even pizza places or grocery stores can end up scaling way beyond the original aim of the founders. McDonalds is a nice example.
If you want a single word that identifies companies that are growing very large, very fast I'd suggest this one: lucky.