Actually, the word rut comes from and Old French word (don't remember correct spelling) which means to roar or grunt.

Old French word "rut" which is translated to "heat"

Found it. Old French - ruit (a roar or grunt)or Latin - ruditus (a loud roaring).

I'm a bit of a history buff...I read a lot of books on the American Revolution and the settling of Kentucky...I remember reading an account of a hunt that Daniel Boone was on and he mentioned that it was during the time that the bucks were "rutting"...

So, it has been used a long time...I bet that the French word would be correct, and we just Americanized it...

People still use "rutting" at least in my parts.

Anyway, my life feels complete now that I have learned the etymology of "rut."