Western hatchets are some sort of small axe, but japanese hatchets (more important for danbooru I guess) look like a big cleaver, a common yandere weapon. A billhook is basically a japanese hatchet with a hook, see Ryuuguu_Rena for that one.
The hatchet currently has no wiki entry, but it sure should get one. I'd use the japanese approach and define it as a big cleaver with a long handle so you can hold it with both hands. Alas the tag is also used for a small axe, the aforementioned western hatchet, so this needs some cleaning.
No big problem, there are only about 20 entries currently. Splitting hatchet into two tags for japanese and western ones probably is the way to go.
Think of it as more of a hatchet than machete. It is carried by woodsmen for cutting/harvesting green bamboo and other finer woodworking purposes and for splitting firewood (batoning anyone?) rather than slashing like with a machete.
I think the name 'hatchet' is based on how people use the tool, although that tool itself is technically a machete.
A cleaver is more of a butcher's knife, a hatchet is an axe used to cut and split wood (specifically from a tree), a billhook is like a sickle yet functions like an axe to cut tree branches and twigs, while an axe is simply a generic chop and slice weapon mainly used in war (take for example, a hand axe and a battle axe).
We can't really differentiate between similar looking objects like a japanese hatchet and a cleaver by usage if we don't see them in use. Like dean exia mentioned both are a specialized form of a machete though, so tagging them that would be a solution.
The word for chopping knife (hatchet) in Japanese is 鉈 "nata". You will get an idea of the variety if you do an image search for that character.
- there are 腰鉈 koshi-nata, meaning "worn at the hip", a general use hatchet (also万能鉈 bannô-nata) - 竹割り鉈 takewari-nata, a nata for splitting bamboo (this one often comes w/o a wooden handle) - 剣鉈 ken-nata, literally "sword-nata", is a clip point nata mostly used by hunters - 嘴付き鉈 kuchibashizuki-nata with a blunt or pointed nose - 鉈鎌 nata-kama, literally hatchet-sickle which is probably closest to the idea of a billhook - 枝打鉈 edauchi-nata which are a kind of hybrid between a hatchet and a handaxe
A lot of nata are chisel-ground for right hand use, except takewari-nata and ken-nata which typically have a standard two bevel grind. There are countless regional variations, too.
Just for reference, Oxford Dictionary definition of hatchet:
a small axe with a short handle for use in one hand.
Middle English: from Old French hachette, diminutive of hache 'axe', from medieval Latin hapia, of Germanic origin
I'm all for splitting off an oriental_hatchet or bamboo_hatchet tag, or similar. As it stands, the posts under hatchet all feature one-handed axes, or what seems to be more like a one-handed fireman's axe in the case of post #1194669. However, the tag should not imply machetes, as machetes were/are substantially different in their construction (machetes are less squarish with elongated blades about 30-50 centimeters long, and are usually slightly curved, tapering to a blunted point; while bamboo hatchets are fairly squarish with no point and a straight edge) and origin (South American versus Far Eastern). Also, as I seem to gather from looking at photos, the oriental bamboo hatchet is about 1,5-2 times the length of a usual meat cleaver, making for a fairly clear differentiation.