I have always had a problem with that. A start-up is a start-up if you decide it's a start-up and want to transform it into a viable business. There is no magic line that you have to cross to transform your project into a start-up. I always advice people to call their projects a start-up from the beginning.
Could you expand on why you advice this, meaning, what is the advantage of it? According to your post simple message above it could be the other way around (always call your startup a project).
Secondly, other people take it more seriously. Everyone has some sort of a side project, very few people have a start-up. They give you more time and listen to your more intently.
All anecdotal, no data to back any of the claims.
(ofcourse, opinions differs)