The term is no longer strict. People and press call companies with 1 billion valuation startups just because they're young and disruptive (AirBnB a couple of years ago). At the same time one-person bootstrapped websites call themselves startups. I've seen a chef call himself startup (instead of simply entpreneur).
I think you're right: once the company has traction, found product-market fit and starts to scale its processes (with external funding or not) it should become just a 'company'.