OP, I actually HAVE read all ten pages to this point. Sounds like a very bad deal for you, and I’m sorry you are hurt, as well. 
Have you worked out on paper the itemized show bill that the trainer gave you, and what you should have been charged? And have you clarified/included any prize winnings? Having all that in black and white, itemized, will be a useful tool that will be hard to argue about when you have your meeting.
I would second the suggestion of having another person there with you when you meet with trainer, as was mentioned earlier. Not because you aren’t capable of having the conversation that needs to happen, but more to have someone there to be YOUR advocate, your supporter. Aaaaand just in case, you have a witness, so the trainer is less likely to try to pull any crap with you.
I also like the idea of direct yes or no questions, because the response, even the non-verbal parts, will tell you a great deal. I would love to be a fly on the wall when you meet, and ask trainer very plainly, yes or no, if she double-billed the second rider, too.
If it helps you organize your thoughts, you may want to plan your conversation in the form of a business meeting agenda.
It might be something like this:
A. Bill for the show and accounting
B. Current lease agreement and terms; what is customary practice in a 100% lease agreement
C. Revising/updating the lease agreement to include items that need to be added
D. Review of discussion, date for follow up meeting
It may help to say that you need to ask her to put on her “Trainer” hat for items 1 and 2, and then change to her “Horse Owner Lessor” hat for the rest.
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Begin with the show bill, show the trainer’s itemized bill, and YOUR itemized bill. Show what you believe to be correct, factoring in the prize winnings.
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Get a solid agreement and commitment on what amount is due back to you, and when you will receive it FROM THE TRAINER. (I would keep that accounting clean and clear–get a check from trainer for the show bill, NOT a discount or decrease in the next month’s lease fee or the like. You can just say that you are keeping your own accounting of your horse expenses and need to keep your records of expenses in correct categories.)
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Move on to the lease agreement as it currently stands, and list out the items in the agreement that support your position as leasing the horse 100%. If the lease does not contain such language, you can take the many suggestions on this thread, and say that your understanding of a 100% lease is X, y and z, as well as 1, 2 and 3. And that you will require a revised lease agreement to contain those terms.
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Add in any further terms that you feel are necessary, such as whether someone else can or cannot ride the horse at ANY time, take horse to shows, etc. This is one place you can indicate YOUR control over what happens with the horse.
If you have been less-than-sure about what a lease agreement normally covers, this is where you can make it known that you DO, in fact, know what is customary in a 100% lease agreement. It will also show that you will not accept any more attempts by the trainer to bullsh*t you! If you loff the horse and have liked the trainer, now is the time to indicate that you wish to continue the lease, orrrrr NOT.
- Review the four points discussed as clarification that both parties are in agreement, and offer to write up a summary of your discussion, and a new lease agreement. You may offer to do so, as a working draft, that you both go over to make a final draft.
Do NOT allow trainer to bullsh*t or steer the conversation into 800 directions. Simply repeat, “Please let me continue on this topic, I want to be sure I cover these important points with you, for clarification that will benefit both of us.” (Try hard not to roll your eyes while saying that.)
These are only suggestions to keep an organized plan for your meeting. Your “meeting agenda” may look different than the examples above. Having it organized, planned out, written out and thought out will be very useful to you to stay on track. It also shows that you are approaching the situation in a professional, unemotional manner, just the same as any other business type meeting. “Wait on that thought please, I want to continue discussing item 2 right now.” If you tend to take notes during a business conversation, bring a notebook and jot down the important bits. And let trainer see you doing that. 
Apologies for the length, I was attempting to include the many comments and suggestions held in the previous ten pages!