I think you came to the wrong conclusions from that episode. Neither one was cheating, unless you consider wings to be inherently cheating, but Applejack spends all her time training her legs, while Rainbow Dash trains her wings. Both of them very evenly started to get angry over nothing, and evenly escalated into a full-on fight. I think the writers went out of their way to show that neither and both of them were at fault for the fight.
RD likes to compete and she likes to win, but she also likes to play and she's willing to lose if she's playing. But she also has her limits and she doesn't like it if she's just playing a game and someone's seriously trying to beat her.
In the first scene of the episode, RD is winning at horseshoes, and AJ picks up RD's shoe and sets it on top of her nose. RD plays along with the silliness and tries to shoot from that position, and can't hit the side of a barn. Then AJ wins from behind and it bugs RD, and AJ starts talking like it was a serious win.
Then when the contests start, RD deliberately holds back and doesn't use her wings, even though nobody ever asked her to do that. She runs the first race without the benefit of her wings and loses, and even admits that AJ is faster, but Spike gives her the win on skill, because AJ took a penalty. AJ goes on to mostly win contests that are slanted in her favor, but RD surprisingly wins enough random skill victories to keep the game tied.
At the halfway point, RD very obviously uses her wings (no-hand push-ups), but AJ says nothing about it. This prompts RD to use her wings again and again, but she's using them in a very taunting fashion. RD is pushing AJ to say something about the wings. She wants AJ to recognize that she was holding back because it's just a game. But AJ won't say anything, because by now she thinks it's a real competition and she thinks she can beat RD, even with her wings (wings for example, wouldn't have helped RD smash the strength meter). AJ kind of has a problem with that. She thinks she can outrace Rainbow Dash. She thinks she can out-trick Trixie. She thinks she can harvest an orchard by herself. She's good, but she doesn't accept her limits.
RD keeps poking away at AJ's bottled emotions until AJ explodes and accuses RD of being a cheater, and they start the final contest/fight in bad faith.