It's a matter of opinion. I've seen some people put the Classical era starting in 510 BCE, when Hippias, the tyrant of Athens and son of Pisistratos, was at last expelled and the form of government there became democracy. This is, of course, very Atheno-centric, but Athens is the city from/about which we find the greatest quantity of written records. I've also seen 490 BCE used - the date of the battle of Marathon, when the Persians attacked the first time while they were ruled by Darius. Then, as you say, 500/499 and 480.

It's not a fixed point, but I wouldn't consider it a serious dispute. It's a boundary used for classification purposes, with each point chosen based upon a significant historical event which had an effect on what follows. It's not really a situation of "before this point, they were Archaic and that means X, Y and Z; and afterward they were Classical and that means A, B and C. As with many things, change came gradually in matters beyond politics, such as day-to-day life, art, architecture, social interactions and so on.

So really, the answer is "whichever works best for your purposes."

It's a rather arbitrary division of time. Most ancient historians place the beginning of the classical period sometime between the invention of Cleisthenaic democracy (c.509 BCE) and the end of the Persian Wars (480/79). The Battle of Marathon (490) is also used as a marking point. I typically tell my students to remember 500-323 as the classical period.

It's important to remember that this is an arbitrary date. It indicates to us how important Athenian culture has been to the development of Western history as the most well documented Greek polis.