The trouble with the word startup is that it means different things to different people. Taken literally, it could mean any company which has just started up. That's not what the word usually means when used on HN, of course. I recall this pg essay http://www.paulgraham.com/startupfunding.html which notes the difference between being a band that aims to sell a million copies of a song versus a band that is content to play at weddings and bar mitzvahs. A hairdresser is more likely to be the latter than the former. It can still be a startup but it's not a "Startup".

I'd still take the position that a hairdresser could be a Startup (as well as a startup). There are a few well known brands that I expect originated in that fashion(!) and there's always room for further innovation.

In our local HN echo-chamber terminology maybe not, but it definitely does count as one in the world that the original government backed site is targeting. In that case, "startup" is synonymous with "new business" and I personally don't disagree at all.

Follow the Inspiring Stories under Getting Inspiration - most are not tech-startup focused. Also look at 'About Us' and read the bios. I'd argue that 'startup' for the government relates to any new small business, not just a tech/web focused one.

An established hairdresser may not be a startup, but I see no reason that a new hairdresser can't be called a startup.