Hi
The original item is almost certainly a plough coulter - the origins of the word for knife in Latin 'cultellus' also gives us words such as cut, cutter in English and couteau in French.
Many agricultural tools could have been used as weapons, or turned into weapons.
Bill hooks can be concave (i.e. with a hook), but also convex or even straight bladed (see
www.billhooks.co.uk), or with a back blade, or a hook.
Fitted to a long handle they become the English slasher, the American brush axe or the French croissant.
Given an even longer handle they become pole arms - various names given, depending upon the blade shape and country of origin - in the UK the most common was the bill....
Billhooks were just known as bills until the 19th century - sometimes with a prefix such as hedging bill - hookbill is another spelling - origins probably hackebeil (DE) or hakbijl (NL) - meaning a chopping sword or axe....
The Dacian falx, one of the few weapons to strike fear into the Roman legions, was shaped like a big billhook. Falx in Latin means sickle, or with a suffix it means billhook (falx vinitoria, falx silvaticus, falx arboraria - spellings may be off at it is late at night here in the UK, and well past my bedtime...)