I think there are two common definitions of startup, and neither require VCs to be involved.
One (seen elsewhere in these comments) is any small business. I personally don't like that definition because there is a pretty big difference between a local coffee shop and the thing we all mean when we say "startup".
The other one which is more common here is a company that is currently small, but the business model involves getting much much larger. There's a blurry line between a small business and a startup with this definition, but it seems to be a "you know it when you see it" type of thing.
Companies like Mailchimp and Atlassian (in their early days) clearly qualified as startups even though they hadn't raised VC. You might say they're outliers, but so are the VC-backed companies that reach that level of success. If a small company is growing quickly and on pace to become a multi-billion dollar company, it seems weird to say they're not a startup just because they didn't raise money from the right people.
I’m by no means an expert on this, but I never assumed “startup” implies VC-backed. I thought “VC-backed startup” meant that and “bootstrapped startup” meant not.
That “startup” was an indicator of not being an incumbent in the space and using technology to disrupt things.
But in an increasingly tech-first world, it does beg the question what separates a tech first SMB from a startup.
And VC’s invest in companies to get a 100x return.
Which almost by definition means, if you run a startup - you need to get it to become a unicorn for it to be successful.
Otherwise, why take VC money … and just bootstrap it instead.