Being publicly traded doesn't automatically mean you're not a startup. Startup = growth, and there are plenty of other things aside from stock price or employee count to look at. For example, Tesla was public when they just made a single car model, but now Tesla is an energy company and expecting rapid, multi-year exponential growth from new markets. That kind of search for product market fit and growth from a company is much more of a startup attribute than anything else, and looking at how a company plans to grow is a much better indicator of how we should use 'startup' than "is it public or private".
http://www.paulgraham.com/growth.html