It's refreshing to hear from someone who's worked for the regionals and has now gotten enough sleep to be able to join a thread full of Phoenix Wright lawyers.
Denied boarding/offloaded after getting on the plane means the same thing in the real world.
I will agree that is how the UA and Republic treated it, and that's probably the mentality that many of the people who have been derided on this thread have. But that is a non-obvious assumption, since the documents posted here make a clear distinction between "denied boarding" and "removed from aircraft", and they have different criteria.
Clearly, it's in the interest of the operators to have "boarded" count for revenue purposes as early as possible (e.g., when the ticket is scanned at the gate, and the machine registers "boarded"), and for denied purposes, as late as possible (e.g., anytime before V1).
Company policy, even "Industry Practice" doesn't automatically make case law, and it's probably in the interest of the whole industry that this distinction be left vague, so that operators can interpret it in their favor.
In the US, to answer a question raised above, the right to a trial by jury is enshrined in the Constitution (thank you Mr. Blackstone) and can be invoked in both criminal and civil cases. In civil cases that are complex and involve two corporate parties, the right is usually waived. In a Joe-Public-vs.-Big-Corporation Tort case, Joe Public often wants his jury. When you hear of massive awards in civil cases (like "1 day's revenue from all the McDonalds in the world), it's often a jury decision (then, on appeal, the whole thing gets knocked down).
The victim here is allegedly a poker player, and a fairly wild one at that. So while the smart move would be to settle quickly for a large sum of money, he and his legal team might be willing to risk the expense of a jury case to get a massive award, which they can then use to negotiate a better settlement in exchange for dropping the appeals.