Do you think your principle ["if you merely participate in a gambling contest with a desire to win, you are guilty of coveting that which belongs to your neighbor"] applies exclusively to gambling, or does it apply to any and all forms of competitive activity? For example, if I enter a boxing match, or any other competitive activity, I am entering it to win. Does this mean that I am guilty of violating the tenth commandment by coveting my opponent's title, belt, or even reputation? It seems to me that it does, but I'm not sure.
Possibly but not necessarily. There's nothing wrong with competing in a contest to win. The apostle Paul clearly commended that desire in 1 Corinthians 9:24-27.
There are undoubtedly times, however, when an athlete's motives might be tainted with sinful pride and even sinful ill-will toward the opponent. (I think that's been a pervasive problem in modern professional sports at least since the time of Cassius Clay.) But I don't think that's always necessarily a part of athletic competition, or Scripture would condemn athletics altogether.
What makes gambling different, and always wrong in my estimation, is that there's no way to win without actually hurting other competitors. Your victory costs them something real (not just their own pride or title), and what you win is something to which you have no legitimate entitlement. Therefore, I have suggested it's tantamount to stealing.
Winning someone else's title isn't nearly the same thing, because you are entitled (by definition) to compete for that. It isn't really the other person's personal and private possession—except for a stint between contests.
I made note of this obliquely in my description of why gambling is tantamount to stealing: "It is the taking of that which belongs to your neighbor and to which you have no right."
You couldn't say that about the title in a sporting contest. You have a right, if you have the ability, to be champion of the US Open. But if you win, that title is rightfully yours for only one year, unless you legitimately win it again.
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