If a character is casting, you will know that they were casting (you can't both silence and still your spells until epic levels) and if the IC rule was "no magic" then you just saw them ICly either use magic words or make funny-looking magic gang signs. If you fail your spellcraft check, you don't know what they were casting, but you still know something weird was going on.
If a character drank a potion, you will see that they drank a strange liquid and magical stuff happened afterwards.
If they activated any other type of magic item, you will have a message along the lines of "Johnny Fighter used an item" (or maybe "Johnny Fighter used Whetstone," I can't remember). You will know an item was used, but it could be a whetstone, which isn't magical. You should infer based on what you see (did anything start glowing? exploding? floating?) You can ask ICly, "hey wait, is that thing magic?" You can ask, ICly, to see the item, and determine for yourself whether or not the item is magical in nature.
I think even a barbarian knows that when something is glowing, sparkling, exploding, or otherwise acting unusually, it's probably magic. A wizard did it. (Or any number of other magical persons...)
Now, let's say something doesn't have tell-tale signs of magic (no VFX). If what you see is what you get, and you don't see anything out of the ordinary, then you can assume nothing is out of the ordinary, unless, ICly, you're paranoid, in which case you have no evidence for your assumptions but that's just even more suspicious, isn't it?
You can wrongly assume that a machine that discharges electricity is magical. Or maybe you'll read a paper saying mechanics/physics is a form or study of magic. Or one that tries to separate magic and science. Perhaps you'll read a book that says the surface is heaven, you are a god, the gods are dead (therefore you are dead), and if you kill yourself you'll escape from Hell.
People only have their experiences to guide them, and you face the risk of being wrong. We want everyone to remain in-character regarding in-character happenings, which is part of why it is inappropriate to try to acquire information in certain OOC ways. I will repeat, this means you face the risk of being wrong. Or not knowing.
So you don't have any proof that guy was cheating, but you're ICly too proud to accept the idea that you lost? Find out if he has any mage friends, pay some shady guy to spy on him. Give him a big bonus if he reports, "yeah, that guy was thanking his sorcerer pal for the Strength of Bulls spell he cast before entering the ring!" (But can you trust your informant? Oh, the scenario never ends! Perhaps why Howl didn't want to list anything. It also might put ideas in your head - ideas which really are more satisfying to come up with on your own!)
My possible thought-hijacking aside, I hope this has helped you see that there's a reason for keeping it IC, and there are possibilities out there to approach even seemingly blurry issues ICly.