"Proper" chilli means really really as authentically real as actually possible. Which presumably means as made by some Spanish pioneers or central American Indians. It was probably very plain as gourmet cooking wasn't particularly big on the New World frontier I imagine.
Just checked Wikipedia:
In Spanish, the "chile" refers to a chile pepper and "carne" means meat. The first documented recipe for "chile con carne" is dated September 2, 1519. The ingredients were boiled tomatoes, salt, chiles and meat. Bernal Diaz del Castillo, one of Hernan Cortez's captains and the source of the recipe, states in his book, that the Cholulan Indians, allied with the Aztecs, were so confident of victory in a battle against the Conquistadors the following day that they had "already prepared cauldrons of tomatoes, salt and chiles" in anticipation of a victory feast. The one missing ingredient, the meat, was to be furnished by the Conquistadors themselves: their own flesh
So a REALLY authentic chilli does contain tomatoes and also human flesh. And probably tastes a bit rubbish, looking at that list. Mind you, human flesh might add that special elusive flavour and make it taste great for all I know.
To the OP - if by green peppers you mean green bell peppers then NO. NO TO PEPPERS IN CHILLI. Not for reasons of authenticity, but because the flavour will sneak into everything and make it taste thin and peppery.
I love peppers, red and green, but not in anything that stews down because their undernote just wipes everything out. They are best enjoyed cooked but distinct.. like in fajitas or quesedillas.. nom...