I'm going to get really repetive here because you just don't seem to understand how things work and in coming capitals just to get my point across.
Azula did LOSE the Agni Kai because if she had won SHE WOULD HAVE BEEN FIRELORD.
Azula DID break the rules because if she didn't she would have won and SHE WOULD HAVE BEEN FIRELORD.
You can't make up the rules to suit your own agenda espeically when everything is pointing to Azula breaking the rules and no one, again not even Azula ever argued against the legitamacy of Zuko being crowned over her in terms of losing or winning the Agni Kai. In fact no one questions it, not even those who was firmly against Zuko even though they could've used that against Zuko and put someone else on the throne. They didn't because they know the rules and they know that Azula cheated. You can try and argue the fact all you want but unless you can provide actual proof, like a writer saying Azula technically won or even a character saying she did then you have no real evidence to go against what is clearly shown in the show only your own biased headcanons.
Katara is being brought up because she's an important factor in Azula cheating. You bringing up a fight that has nothing to do with the Agni Kai in question is not an important factor in discussing why Azula lost.
@EmBELLEm
Lol that actually made me laugh :D
If attacking bystanders were allowed in the rules I'm surprised at how many are willing to get in the way of possible danger. Either the fire nation are complete idiots or they just don't have any common sense.