The difference between a personal project and a startup is that a startup has a primary goal of profit, whereas the personal project may just be for fun or a personal exercise.
Only on HN, I hope. A startup is a business, a corporation, with the intent of making money. A website is, at best, a product, possibly of that start-up.
Is a buying a corner-store or laundry mat mean you are running a startup? I think there are several other qualifications needed.
1.) There needs to be the intention to hire more than yourself/family as employees.
2.) There needs to be the intention to pay a return on investment to investors.
3.) The organization needs to be structured as corporation.
Plenty of start-ups postpone the incorporation step until they have sufficient traction. There are good reasons to do so, for one during the initial phase the 'founders roster' is subject to change and once someone is on board it can be very hard to remove them again if they have stock. So it's better to wait for a bit until the 'shake-out' has taken place.
That way you have a much better idea about the amount of weight each of the founders is going to pull and the stock issued can reflect that.
Before the real work starts everybody is always available, once it starts you know how solid those commitments are.