According to Websters Dictionary, you're both wrong. This is the entry for Chili Dog:
Main Entry: chili dog
Function: noun
: a hot dog topped with chili
And there is no entry for Coney Dog. But there is this entry for Chili Con Carne:
Main Entry: chili con car·ne
Pronunciation: "chi-lE-"kän-'kär-nE, -k&n-
Function: noun
Etymology: American Spanish chile con carne chili with meat
: a spiced stew of ground beef and minced chilies or chili powder usually with beans
Therefore, they are both chili dogs.
Main Entry: chili dog
Function: noun
: a hot dog topped with chili
And there is no entry for Coney Dog. But there is this entry for Chili Con Carne:
Main Entry: chili con car·ne
Pronunciation: "chi-lE-"kän-'kär-nE, -k&n-
Function: noun
Etymology: American Spanish chile con carne chili with meat
: a spiced stew of ground beef and minced chilies or chili powder usually with beans
Therefore, they are both chili dogs.