The one thing they all agree upon is that it came from Texas! Real Texas chili, no beans!
Extract from an 1877 Kansas newspaper:
at San Antonio they have a dish called chili con carne. It is of Mexican origin, and is composed of beef, peas, gravy and red pepper.
Another tidbit:
In describing San Antonio’s open-air chili scene in 1882, a reporter from Alabama’s Greenville Advocate rendered it as “chille cancarne.” (And also reported that it contained beans.)
It wouldn't surprise me if no beans in Texas chili is a recent invention, much like Santa's suit being Coca-Cola red.