yes. Axrendale explained it perfectly.

even if you let her go doesnt mean you dont love her anymore

Basically, in order to answer this question, we need to define exactly what "attachment" means. At no point did the guru ever say that Aang needed to stop loving Katara - he just had to learn to let go of his attachment to her. If attachment can be satisfactorily defined in a manner where a person who has let go of it can still be in love, then the argument is over. Since Avatar has been decidedly vague on this point (no doubt intentionally), we are forced to turn to an alternative source of wisdom; a franchise that has more experience in the ways of "attachments" than you can poke a stick at; the source of all my comparisons to the movies and the spin-off series: Star Wars.

Anyone who has watched the Star Wars movies is familiar with the fact that the jedi too seem to have a bit of a problem with "attachment to people" (what with it leading to the dark side and all). What not so many people are aware of is that the jedi do in fact find a solution to this problem in the series of novels that tell the story of what happens after "Return of the Jedi". Here is a direct quote from one of the novels:

"That's what attachment is, isn't it? It's not loving somebody. It's not marrying somebody. It's not having kids. It's being where, if somthing goes wrong, there's nothing left of you. It's being where, if the person you love dies, you start functioning like a droid that's been fitted with a restraining bolt".

I think that this definition could easily apply to avatar. Letting go of attachment doesn't mean loving what you were attached to any the less. It means that if you have to make a choice between saving the person you love and saving the world, you have to be able to accept the pain of letting go of your love in order to save the world. After that, you can mourn the person you loved, but you have to be able to keep on living and get on with your life. You can feel the attachment, but you cannot allow it to have any hold over you.

What evidence do I have that this is the right definition? Simple; it fits the facts.

Let us consider the scenarios we have seen so far. In "The Avatar and the Firelord, Avatar Roku perfectly displayed this ability to "let go" of attachment. When the volcano errupted, his first instinct was to get his wife to safety. But he didn't. As much as he wanted to put her first, he knew that his duty as the Avatar was to hold back the lava until the entire village had been evacuated, and so he let go of her hand, and went back to make his stand.

Aang didn't need to stop loving Katara; he just needed to learn to put the world before her. In CoD, Aang was forced to abandon his urge to protect Katara so that he could summon the Avatar state. he had to take the risk that she wouldn't survive in order to do his duty. When we get to "The Awakening", Aang shows this ability again, when despite his obvious continued feelings for Katara he leaves her in order to do his duty.

There. That's my opinion on what Bryke have tried to set up for us. Before anyone complains that it wasn't very well explained, I would like to say that I love series that force their fans to figure things out for themselves. It keeps things interesting.

True, the actual specifics of the rules of an Agni Kai, and whether or not Azula would have obeyed them remain somewhat unclear. But still, the point is that (as you said), Zuko could have taken Azula out while she was casting the lightning at Katara, but he didn't. Instead he almost died saving a girl who ony a month before had been promising to kill him. I think that given the choie between displaying the improvement Zuko has made with his firebending, andthe improvements he has made with his character, I would have wanted to see the improvements in his character.