Let's assume there are two players - Player A and Player B.
1) Let's assume Player B cheats - in this instance only Player A can play a cheatin' resolution as the rules state the cheating player (the person who reveals a cheating hand) cannot trigger their own cheatin' resolution based on their own cheating.
When a cheatin' resolution is played against someone, what it means is that the cheatin' resolution is being played against another player's instance of cheating (not necessarily the card effect). So if Player B cheats, Player A can play a cheatin' resolution against them. Sometimes cheatin' resolutions do not affect the cheatin' player (One Good Turn as an example) but are considered played against them. The reason for this quirk is that Player A can only trigger a cheating resolution off of Player B (otherwise, Player A could cheat and target Player A!)
There are currently two instance when cheatin' resolutions are relevant.
1) During shootouts, if a player reveals a cheatin' hand, there is the resolutions phase where players can use Resolution abilities and Cheatin' Resolution abilities IF the opposing player played a cheating hand. The round winner (the person who won lowball) gets the first attempt to play a resolution. So for example, Let's say Player B cheats during a shootout, by revealing a cheating hand (and let's assume Player A won lowball), the following could occur
Action 1 - Player A plays Point Blank
Action 2 - Player B plays Hex Slingin'
Action 3 - Player A plays Coachwhip!
Action 4 - Player B passes as they have no more relevant actions
Action 5 - Player A passes as they have no more relevant actions
The playing of the Coachwhip! is being played against the instance of Player B cheating. The same could be applied if it was Magical Distraction or Fetch. It's always against the instance of a cheatin' hand (even if the effects of the Cheatin' Resolution card don't target the actual cheater!)
2) Cheatin' Resolutions (but not normal Resolutions) can also be played during lowball. The person who gets the first opportunity is the previous day (aka turn) winner of lowball. In the event it's the first day of the tgame, you may pull the top card of your deck, and the person who pulls a lower value gets the first opportunity to use a cheatin' resolution against the instance of cheating.
The most recent rulebook can be found here (The base set rulebook has been updated):
Apologies if that was confusing!