Playing monopoly with friends and kickball at recess are two very different situations. Monopoly has a set of written rules that people can refer to, kickball rules are much more negotiable. Usually, even the dimensions of the field are arbitrarily set by the players themselves: first base is the patch of brown grass, second base is Jamie's sweatshirt, etc.
I've worked as a school staffmember during many a recess, and it's true we only intervene to keep students from hurting themselves or others, or to occasionally settle an irreconcilable dispute. We do this so the children can learn valuable social skills on the playground for themselves. And for the most part, they do a very good job of playing fairly. Recesses are short and kids learn that games progress more quickly and are more fun when there are agreed upon rules and everyone follows them. Peer pressure usually keeps kids from cheating. Sometimes rules are applied differently to different people, to accomodate for different ages or children with disabilities. I think this is all fine as long as everyone is in agreement.
Sometimes, however, things can skew the situation. Older kids or the popular crowd might use their extra influence to bend the rules to their unfair advantage. Again, this is a learning experience for children, they have to decide whether to go with what the in crowd wants, stand up for themselves or go play something else.
I think instead of the issue being about whether you're a cheater or a stickler, it deals with social and power dynamics. If it were my son, I'd tell him to stick up for what he believes is right and fair, but not to impose his beliefs on other people who are in disagreement. Sometimes unforseen situations arise in a game requiring an improvised rule, or there are genuine disagreements with what people see as fair. In the situation in the article, it seems that the fall down rule is an attempt by the other team to manipulate the game to their favor. It shouldn't be very hard for Eli to convince his peers that it's a stupid rule. If Eli's team and his opponents can't reach a workable compromise, maybe there's another ball and a space where he and his friends can have their own kickball game, leaving the floppers to theirs. If everyone else likes the fall down rule and agrees to play with it from that moment forward, Eli can either go along with the rest of the group or head on over to the tetherball pole. Either way, it should be his own decision