Tan the color is a bit later, mid seventeenth century. It’s the color of tanned leather. Earlier cites are to the tanning process, not the color. Beige and taupe both come from French, imports from the fashion industry. Khaki is from the British army in India and is from Urdu. The “not cute enough” theory works for beige and taupe, but not for khaki, which has a different route into the language.
I’m not sure these have widely accepted definitions of difference. I suppose Pantone might set a standard, but I don’t know if they use these words or even how widely used the Pantone standards are. (Not my area of expertise, plus I have green-brown color blindness, so I’m not a good judge of subtle differences in browns.)