I've always thought of it as a service, since it accepts payment and gives you something in return. What exactly is a startup? I've worked at a couple -- I kind of think it's a marketing term. And well, almost certainly means pre-IPO.
A startup in regular English is a recently begun business. It doesn't have to be tech related or doing anything spectacularly new, it could be a family run bakery in a small town.
A startup in HN English is a recently begun entrepreneurial business. It's trying to do something new and innovative and usually aims either to be acquired by a larger company or make absurdly large amounts of money and IPO.
A startup is a company that hasn't found or proven a reproducible and sustainable business model yet.
Basically a startup is a company that hasn't yet found a way to reliably bring in enough money to operate. I think this definition can also apply to non-tech startups like a grocery store.
Website is something which is showing the information of an entity it can be a nonprofit organization or a business, so a website is not a business in itself as a startup. I checked your link of BetaPage and I think it is more like a startup
A startup is a business. Pebble was a hardware startup that had a website. Facebook was a software startup that offered a social networking website.
As for what exactly a startup is, look no further than PG's own description: http://www.paulgraham.com/growth.html
A startup is a business that arbitrages a temporary economic disequilibrium to produce extraordinary growth.
Most startups have websites, some startups are entirely web based, but most websites have nothing to do with startups.