A helm is different than a helmet and it isn't short for a helmet. Its more accurate to call it a type of helmet, or if you want to, say that a helmet is a type of helm.
A Helm refers to the old Medieval style of helmet, if nothing else, but it is visually and functionally distinctive from the modern helmet.
As Anelaid notes, the two terms have different meanings. There definitely needs to be a distinction written into the respective wiki pages.
However, I wouldn't disagree with a helm → helmet implication. A helm is still protective headgear, even if it's medieval-style. When I think of the headgear on a suit of armor, "helmet" is the word that springs to mind.
Hey, this is Danbooru. Arbitrarily narrowing definitions to improve index-searching is what we do.
edit: Speaking of definitions, I'd like to slightly broaden the helm definition. I consider the Dragon Quest 3 female soldier's helmet, and those like it, to fall under that category.
I looked it up in the encyclopedia and got something different than Wikipedia, and honestly, I trust that over Wikipedia, especially since the Wikipedia page is a disambiguation. I also don't know if danbooru has a stance on using Wikipedia to back up things like definitions.
I've also cleaned up the definition of helm to be more accurate as to what a helm actually is.
葉月 said: You can always edit wikipedia, y'know. And not trusting their articles until you've checked them is a part of balanced breakfast.
Well, I edited Wikipedia then to be more accurate. It seems like their main page is focusing only on the modern helmet so that explains why the error occurred.
If the lamp isn't present, then it would look no different from what would be tagged a hardhat. So the only thing to really distinguish a mining helmet from a regular hardhat is the presence of a lamp. The lamp need not be built it, but it certainly needs to be attached.