Over mixing, when do I have to be aware of it? (Baking cookies)

I don't think that your problem has anything to do with overmixing.

1. With melted butter, it doesn't really matter once it looks reasonably homogenous. With soft (22-ish Celsius) butter, longer mixing is even better.
2. You cannot overbeat an egg for cookies, as long as you are using whole eggs. it is possible to overbeat egg whites only, or omelette mixture, etc., but those are different cases.
3. This is the step about which you hear warnings for cookie making. Beating dough is similar to kneading it, because the wet flour starts forming gluten, which makes the dough more bread-like. So, should you pay attention here? Generally yes, it is a good idea. But in practice, it is pretty hard to overbeat cookie dough at this stage. Bread dough needs several minutes in a strong mixer before starting to exhibit significant gluten creation. The butter in cookie dough inhibits gluten formation, so you could expect to have to beat it even more before you get gluten. Your ingredients usually come together after maybe one minute of beating; you would have to continue beating for at least 3-4 minutes longer before something happens, so beat it 5 times longer than necessary.

And even if you overbeat it, the result is that you get gluten, which makes the dough tough and chewy. So, while this is generally a good advice for most kinds of cookies, a little gluten production will probably not give you much trouble, because you want a chewy cookie anyway. So, altogether, you have a pretty large margin of error. And if you indeed happen to overbeat a little bit, it won't be so tragic. It makes the difference between good cookies and great cookies; it doesn't make the difference between just-edible and good cookies.

As for why your dough is different than the one in the video, I have two suspicions.

First, you may have overheated your butter. You should just melt it on a low flame. If you leave it on the heat for too long, or use too high a heat, you will get browned butter, which is darker in color, and has much less water (the normally present water evaporates). This may account for both the color and the non-stickiness of the dough.

Second, I don't know how you measured. Around here, the US system is practically unknown, and whoever reads a US recipe with cups, usually starts measuring with the first best drinking cup he has lying around. But even if you used a measuring cup, it is likely that you got a large measurement error, because measuring by volume is inherently inaccurate. Too little flour in relation to the butter will produce the kind of dough you are talking about.

And in the end, you don't mention anything about the baked cookies. I have made chewy-cookie recipes before, and they frequently made a dough similar to what you describe: dark and non-sticky. The resulting cookies were good after baking. So, if you posted the question before baking the cookies, there is a chance that the result will be good. Even if not, the problem is likely not in overbeating.