My definition of startup - Startup is what every person who have any idea of it wants to do.
Paul Graham's - A startup is a company designed to grow fast. Being newly founded does not in itself make a company a startup. Nor is it necessary for a startup to work on technology, or take venture funding, or have some sort of "exit." The only essential thing is growth. Everything else we associate with startups follows from growth.
Now we have lot of differences in our definition, but in my view they relate to each other. You see humans crave for fame and they get to know startup can be the key to that. So they wants to do that. So they try one(if they have guts) and wants to get their fame as soon as possible. And how they would get this fame? Through very high growth. Paul give an example that a Barber Shop is not a startup. Yes it is not. Because he is not their to earn fame but earn some decent money (well, most of them). But most of the people who fell into startup fever are from my definition and they think they do bunch of things and create a product and ask their some to use it and that's it. They are the next Mark Zuckerberg. Well from personnel experience, I am telling you, startups are hard, damn hard. You got to give your everything to them.
Now I would not go on my personal theory of startups but you should read that article by Paul. He says two things about startups:
(a) make something lots of people want, and (b) reach and serve all those people.
That's is exactly why startups exist.