I think the criterion I like best is that if you're actively sacrificing profitability for growth, you're a startup.
Obviously there are exceptions in both directions, and other factors to consider, but this seems to be a pretty good starting point for discussion to me.
Unfortunately, declaring your company a start up is sexy. My last company was about 130 employees, made some profit but mostly enough to keep the business afloat, been around for a little over 10 years, and was reluctant to call itself a start up.
The company I'm currently at has close to 1,200 employees, been around for close to 10 years, its own free cafeteria, profitable enough to fund several high-risk initiatives, but easily calls itself a start up..
My personal opinion: you're no longer a start up if you're able to afford your own "free food cafeteria". ;-)
When you have a repeatable business model. Uber has repeat customers but is a long way from having figured out how to make the profits required by their business and funding history.
I suppose that's true, but startups generally take VC funding, and VCs want the company to either IPO or get bought.
There are bootstrapped "startups", but if you're not taking VC funding and trying to go big, then it's less obvious what the difference is between a "startup" and a "company".
Obviously there are exceptions in both directions, and other factors to consider, but this seems to be a pretty good starting point for discussion to me.