From my short research on google:
화재: moreso refers to a disaster and is more often used on the news. Perhaps a fire that destorys a city block
불: general fire, not as specific as 화재
This could be wrong as im no expert, but thats what i found
불 is a pure Korean meaning fire
화재 is a Sino Korean word and it is used to refer a proper incident involving or caused by fire.
화재 = 화(fire, 火) + 재(災, calamity, disaster) = 화마 (somewhat poetic word often used in newspaper headline)
If you are smoking and some Korean ask you like "불 있어요?" he or she means "Got a light?"
You're right, it's what romance languages call an incendio/incendie, an uncontrolled and destructive fire.
In English it's a conflagration, but most people just say fire, which creates this confusion.
We would readily say blaze when we want to specify, I think. (at least in the US) A conflagration is obviously a group of fires, like a gaggle of geese or school of sushi. Also, a word I never hear anyone use.
I'm still (after many years of linguistics and looking up words for fun) trying to figure out the difference between a fire and a bonfire. Bonfires seem to be vaguely larger than a small fire but not a very large fire and it's for staying warm or maybe just a book burning or possibly something else inspecific.
According to Merriam Webster dictionary there are two ideas: The word is derived from Middle English bonefire, meaning literally "a fire of bones." Or, the other theory being: in French, bon means "good," which has lead some to believe that it is the first element of the English word bonfire— "a fire made for some publick cause of triumph or exaltation," and derived the word from the French bon and the English word fire