Outside of the 2-letter acronym world (YC/VC/SV/HN), absolutely entrepreneurs should set their own speed -- they only have to answer to themselves.
This is reasonable though, right? We already have words for such ventures and venturers when they don't necessarily grow fast: "small business" and "entrepreneur". The reason "startup" has entered the lexicon is because there seems to be something categorically different about a large fraction of tech small businesses these days, and rapid growth seems to be a the correct differentiator.
What does one call a new company formed to develop a pharmaceutical candidate? I would call it a startup, but I don't know that such a company has much at all in common with what people around these parts call a startup.
Having a brand new thing to offer the world often entails rapid growth (and the VC funding model demands it), but the idea that growth is synonymous with startups is peculiar to this here community.
A startup is a new business.
There was a brief push to qualify "startup" with "scalable", but the only people that seem invested in this (bogus) controversy are on HN, and HN just assumes "startup" means "scalable tech startup".
New businesses started up all the time over the past century, and there was never a need for additional terminology. People just said "I joined a new business" or "Bob started a small company". Why, then, did a new word "startup" develop? I'd say it's because people intuitively realized there was something about these businesses, which were largely tech and especially software, that set them apart from the existing idea of a new business. When someone starts a landscaping company today, no one ever calls it "a startup", and there's a reason.
Of course you can have slow growing or non growing profitable businesses too. It's a bit of a terminology thing.
so you were close