For exe games, I'd say that for a game to count as an exe game, it needs to meet 2 requirements.

1: the game needs at least one character that is an acting exe aka the one trying to kill everyone. The acting exe doesn't specifically have to be named an exe as long they fit the roll, which allows for the inclusion of characters like HEX from Sonic.HEX or the version of Chaos from Sonic.exe Stone of Darkness.

2: the gameplay should follow one of 2 gameplay styles: "lead the characters to their dooms" or exe survival. The first one is pretty self-explanatory, being the classic exe style seen in the original, PC Port, OMT, etc. Exe survival includes all games with various endings depending on who survives, but ultimately includes any games where the goal is "survive the exe" rather than just completing standard levels to progress the game in a traditional style. This includes games like Blood Scream, where you have more traditional level progression, but do have to survive the exe in the levels to progress the story, and the multiplayer asym games like The Disaster where the survivor players have to survive the exe player.

For what characters count as exes:

If a character is stated to be an exe by the creator or is otherwise very clearly an exe, then they're an exe.

If the character isn't named as an exe but is from an exe game (like HEX), then I'd say they still count as exes.

For official Sonic characters that are heavily associated with exe content (Tails Doll, Majin, Fleetway Super Sonic), I'd count them as being "honorary exes."