It doesn't matter whether it originated in the valley. It's not like calling an elevator a lift. A startup is a technical term with a precise meaning. You can call your bar a startup if you want, but that doesn't make it true.
My point is that it's really not. The term existed before the valley-definition and it exists outside of the valley-definition. Yes, Paul Graham and others popularised that definition (and I personally think it's a good definition and usually use t as that), but it has existed and continues to exist with other meanings, so it's far from a "precise technical term".
If a business is small and stays small, then at best it's a failed startup. If it's designed to stay small, then it's not a startup at all.
You can argue any other point, but you can't avoid the growth criteria. And if you have that, either you grow quickly or your competitor will put you out of business.
My point is that a podcast hosting company has no growth potential. The target market is too small. A startup could use podcast hosting as a stepping stone toward a bigger market, but in absence of that, it's a business like any other. And there's no shame in admitting that.
But it's really strange to insist it's a startup when it's not. Imagine insisting that a desktop computer is a chair just because you can sit on it.
I think companies that don't follow that definition have an agenda: startups are now prestigious, so they call themselves a startup in order to seem impressive.
Sad, but it's just human nature.