Well,it sems that we use only the word Kirves,atleast i dont recall immediately other words.I know that you guys have ax,axe,hatchet,but we dotn.We dont have word for small axe,like you have the "hatchet" and so on..But these could make sense:
Retkikirves = camping axe,usually something you might see as hatchet,but still it can be normal heavy blade but has just shorter handle.
halkaisukirves = splittiing axe
piilukirves (veistokirves / vuolukirves) = axe with blade suitable for shaping,not so much for cutting into pieces.These were used by log house builders in the past.many variants for different requirements but usually these have broad blade in common,but blade profles vary widely,depending on what was it meant for,for accurate or coarse shaping of logs,and so on.
Butchers had their own axes but i have to look out for names of those.
so,i could say that we only add some word,to describe the task that the axe is meant for,with the word kirves itself,like halkaisukirves has wodr "halkaisu" that means Splitting,etc.