New York (city) native here. Usage of "New York City" for disambiguation by Americans, at least, became increasingly common beginning as far back as the late 19th century. But by the time I was a teenager (1980s) I would say that using "New York City" was ordinarily a reverse shibboleth for not being from New York, all the more so to write "NYC". (Where "New York City" tended to be used was governmental contexts, although "City of New York" is the more proper name of the municipality.)

At the same time, the older generation, such as my parents, used "New York" (or "the city") to mean approximately "Manhattan", a holdover from what the city was before the consolidation with what are now the remaining boroughs. That always seemed old-fashioned to me. I believe you can still see some old subway signs that use "New York" to mean "Manhattan". (Incidentally New York is also the name of the county that is almost but not quite coterminous with the borough of Manhattan.)

When I was growing up, the state was more or less never referred to as simply "New York" by people in the New York the city. It was "New York State", most of which was "Upstate New York" (though that couldn't refer to Long Island, and it was never quite clear where 'Upstate' began).

The reverse shibboleth quality of "New York City"/"NYC" is probably dying out, may be already dead. Nonetheless I find it annoying.

It's commonly used as shorthand, yes, but jrockway is commenting in response to a startup program that does include Buffalo, saying not to start your business in New York. I think pragone, responding in turn, was just pointing out that "don't start your business in New York [the city]" is not advice that's incompatible with taking advantage of the Start-Up NY program.

Plenty of times I have mentioned being from Rochester, NY when in Florida or California. People there often assume it is a part of or suburb of NYC. Even after I have told them I am a 7 hour drive from the city and never even visited it until after I moved to Washington DC at 26.

Wikipedia even notes the differentiation is important.

"The city is referred to as New York City or the City of New York to distinguish it from the State of New York, of which it is a part"