What I legally put in my book doesn't affect your game at all.(as there are no elemental school only cards at this time) FAQ page 15 has no meaning to an opponent except if a illegal play is made. In a tournament setting that would be weeded out before it started with registration. Casual play is a little different though we use the honor system.
But it does affect game play. I have to be able to tell if you are cheating or not when ever you cast a spell or take control of a spell, both of which are affected by this rule and would not necessarily be weeded out by registration. If you cast Steal Equipment and take my mage wand, what happens if the spell I have bound to it is restricted by training? Your book was approved as legal by the judges, but now you have the opportunity to cast a spell that wasn't in your book and I have no way to tell if you are cheating or not because you won't give me that information.
The way I understand it, there are two arguments that you might be making. Feel free to correct me if I'm wrong.
Argument 1) Players are allowed to choose to ignore rules or mechanics that they don't find relevant.
We are all guilty of this to a degree. It often facilitates quicker game play and can occasionally just slip our mind. I know that creature has no way to avoid my attack, so I don't need to bother checking before I roll dice. Surely my opponent knows that I'm going to use the Hydra's Full Attack and not his Quick Attack, so I don't have to announce which one I am using.
Even though this is normally innocent, it is not right. A proper game requires everyone to follow all of the rules, mechanics, and procedures at all times. It doesn't matter if you agree with them or not. It doesn't matter if you believe they are relevant or not. If we allowed players to start ignoring rules on a whim, we would have anarchy. If we allowed them to ignore rules when everyone agrees its not relevant, we would have a house-variant of the game and not the official game as written.
So, you might be taking the position that, with the example of the Mage Wand given above, you do not have to reveal your training when using my Wand of of Drain Life because you can not possibly have training in the Dark School. My position however, is that it does not matter if you find that information to be relevant or not. It doesn't even matter if I agree with you. You must follow the rules at all times.
Argument 2) The rules do not require you to reveal information that affects game play.
This argument has a bit more merit. You are correct, there is no place in the rules that says you have to reveal your specific training. You know what else is not required by the rules? No place in the rules does it say I have to reveal my spell to you when I cast it.
By the rules, there is nothing stopping me from taking my face down spell that I've prepared and simply announcing "I'm going to cast Fireball on your mage" without showing you the spell. The rules only require me to announce the spell and the target, which I have done. If you feel I am cheating you could call a judge over to verify that my spell is indeed a Fireball and your mage is in range, but there's no place that says I have to show you the card or tell you anything about it or what it does. The only information you get is the name of the spell and what it is targeting. I don't even have to tell you that it is an Attack Spell, so you have no idea whether your Nullify will work against it or not.
So, while it is an assumption that any information that is needed to verify legal game play is also required to be public knowledge, I think it is an assumption that we can agree upon. Without it, the game could possibly become unplayable.
ConclusionIn the end, my findings are that any information that is needed to make legal game play is required to be public information. Page 15 of the FAQ states that your mage's training does affect legal game play. There might not currently be any cards in the game that make this relevant, but it is not up to the players to decide what is and is not relevant. If the FAQ says that a mage's training affects game play, and any information that affects game play is by necessity public information, then a mage's training is logically public information.