Ok, first thing, the user Enimonemoi from your reignofgaming link doesn't know what the heck he is talking about, the second poster got it absolutely right.
Second: A lot of what people were saying in you forum link was very loosely accurate. It wasn't entirely wrong, but I can certainly see where your confusion stems from
Basically, they are saying the same thing, AAs function like this: Attack declare (you click on your target)>attack delay (number X seconds)>attack hit>Follow through animation (number Y seconds). That is one full auto attack. AS shortens both X and Y, but I do not know if the amount is equally distributed between them, or if it is a champion to champion thing.
Attack delay is what people reference when they say someone has a bad AA animation (like Poppy) or a good one (like jinx), and canceling an auto attack is much easier with a long attack delay, effectively allowing you to fake AA something and then stop it from dealing damage by moving or doing something else. Attack hit is the instance in which damage is applied, and follow through animation or "wind-down lag" as some of the people in your links put it is the part people reference when talking about animation canceling, which is especially important when kiting.
AS per level functions exactly like bonus AS from items, giving you +X% of your base AS as additional attack speed.
Now, I'll use some examples. Poppy has a horrendously long attack delay, but average attack speed. This means her attack hit is much slower than most other champions, even at high AS, but she has the same hits/second as anyone else with the same AS. Basically, the time between right clicking on her target and actually applying damage is very long (almost the full AA animation). This makes Poppy very weak early game because it is easy for enemies to counter attack her while she winds up to attack, and also has hard time doing AA resets with her Q
Jinx has a shorter attack delay, but a longer followup animation. This means the she is much better at canceling the wind-down, allowing her to kite effectively. It is also very hard to cancel an auto attack as Jinx, which makes faking out an opponent harder, but makes lane trading easier.
AS does make it easier to kite, to a certain point. If you have too much AS, your mechanics often become too sloppy to kite effectively since the window for wind-down canceling shrinks as your AS increases, meaning you have to be more precise in your timing when you have more AS, not the AS makes kiting weaker innately.
