Using one name for a bunch of similar items (and giving them the same in-game stats) definitely works. I was watching a television program last night called
Edwardian Farm. On this episode the local blacksmith forged a bill hook for use in maintaining the hedges. This specific bill hook was the Devon Bill Hook, and it was contrasted with the Cornish Bill Hook and while they basically do the same thing in the same way, there's a lot of variety.
Here's a Wikipedia Picture of some of the variety.
Having 21 versions of a Bill Hook in your game is probably excessive. On the other hand, a bill hook is different enough from a sickle or a machete that we don't just call one the other. Mechanically,
as weapons they might be the same. On the other hand, if you want to tie some advantage or cultural association to a specific weapon or variant, giving it a unique name and a different mechanic makes sense. A kukri is a type of knife, but it has an enhanced threat range in D&D. Whether that makes sense or not in the real world case almost doesn't matter. As long as it makes sense in the game (and since critical hits are a little abstract, it probably does) it can be a good thing. Ultimately, what you want to avoid is one version of a weapon being entirely superior so that everyone uses it, or, for most games, making every weapon identical so there's no difference at all between them.
I don't want to leave that last statement unexamined - it is perfectly defensible to say something like 'a hit does 1 damage' whether that's a dagger or a great sword. Or small blades do 1 damage and large blades do 2 damage. But since D&D is largely a game of imagination, I think that it's important to try to provide springboards for the imagination. I think having different weapons with different mechanics encourages that to some degree.