Merriam-Websters Online Dictionary gives a date of 1899 for the origin of this term (which suggests that it might have been coined during
England's Boer War, McTag. However, it give no further attribution. The Online Etymology Dictionary has no listing for that word, either separately or under the entry for fire. Dictionary-dot-com gives a date of 1895-1900, again without further attribution, and lists the Random House Unabridged Dictionary as its source.
I can't understand your objection. Firefight has a very specific meaning, in that it refers only to the exchange of gunfire by small arms units--therefore, it specifically excludes combats in which there is mortar fire, artillery fire or close air support, as well as excluding combat in which armored units or heavy weapons units are involved.