Most people define AI cheat as things the AI is allowed to do by it's programming that would be outright illegal for a human to do.
This is distinct from handicaps where the AI gets bonuses.
Mixed is the case in which the AI continues to get a larger handicap than the human would on the lowest difficulty level.
Civ 1 example of AI cheat: AI periodically granted a wonder
Civ 2 example of AI cheat: AI can irrigate all tiles, even those without access to water. (Human could automate a worker right over an improved tile they wanted to irrigate to take advantage of the same cheat)
Civ 3 example of cheat (that backfired): AI knew where all your units were on the map. It backfired because you could rope a dope the AI by removing all units from the city you wanted the AI to go after and then return the units there when the stack got close and remove units from another city.
Example of AI getting large handicap larger than humans on lowest difficulty level: Civ 4 Unit supply, Unit upgrade, AI Inflation, and AI War Werriousness. First two in particular where the AI paid 30% of normal cost to upgrade its units & 35% of normal cost for unit supply.
There's also "human cheats". Things the human can do but the AI doesn't know how to do at all even at highest difficulty level. In Civ III that was micro manage the science slider to prevent breaker loss and micro manage tile production to prevent shield wastage.