According to the US Small Business Association, it varies depending upon the industry. Here's what they have on Service industries (though I realize many web businesses aren't really service businesses):
> Most common: $7 million > Computer programming, data processing and systems design: $25 million > Engineering and architectural services and a few other industries have different size standards. > The highest annual-receipts size standard in any service industry: $35.5 million > Research and development and environmental remediation services: the only service industries with size standards stated in number of employees
http://www.sba.gov/contractingopportunities/officials/size/s...
There is no hard-and-fast rule or time.
IMO, there is some factor or ratio to where you are now vs. where you want to get to and a matter of risk and acceleration involved in doing so.
EG: if you're selling website templates and making $100K/year and want to make $250K/year and can do so mostly through basic growth with minimal risk, you're not a "startup".
If you want to get to $25M/year and you're going to take some huge risks and leaps of faith to do so at a rapid pace, you are a startup.