Do about peeling wok seasoning

Check the bottom of the pan; often useful information is stamped there. Like the brand, sometimes even model number. Assuming nothing useful:

Steel and iron are ferromagnetic. That is to say, a magnet will stick to them. Both are unlikely to be Teflon-coated (edit: though Wikipedia informs me they exist). The black (which is hopefully somewhat shiny) is seasoning (cooked on oil). Iron would probably mean cast iron, and would generally be heavy (my cast iron wok weighs in at well over 10lbs). So, if they magnet sticks, you probably have steel.

Some stainless steels are ferromagnetic, but most aren't. Stainless is also unlikely to Teflon-coated. But it'll probably be shiny, at least on the outside. So, if not magnetic, its probably aluminum. Aluminum is probably coated.

Teflon isn't the only coating that may be used. It could be anodized aluminum as well (which, I guess, may not technically be a coating). Or one of the newer nonstick coatings. They all look different: Teflon is very dark gray to black, anodized is light gray to dark gray. The newer ones unfortunately are numerous and some can be any color. (Your description of the color sounds like Teflon to me).

You could also try cooking a scrambled egg in it; start it in a cold pan. Teflon in at all good shape will release the egg very easily. Seasoning will, unless very good, probably have you cursing.

You have overcooked the seasoning. I have done this once or twice too. Especially smooth surfaces (e.g. carbon steel) are very prone to this problem, unlike rough cast iron.

What you want is not a dark layer. The layer will darken with time and start looking like usual. But on a freshly seasoned metal utensil, the layer should be yellow-brownish. The stove may be too hot for this, especially a gas stove, I do my pans in the oven, for maybe an hour at 200 Celsius or somewhat lower. A burner under thin metal can cause hot spots of much higher local temperature, where the oil chars instead of polymerising.

There is also the type of oil you use. Generally, unsaturated oils polymerise easiest, but the final layer stays somewhat sticky. Saturated fat can give you some more trouble, but will have a smoother finish. I do multiple layers, starting with 1-2 layers of flaxseed oil - this gives a good basis, it is so unsaturated it can practically dry out by itself in the air - and finishing with cocos fat or lard, again 1 or 2 layers. For the first few time in a new pan, I try to fry with saturated fat too, for your wok you will probably find that the taste of cocos fat has a good harmony with Asian dishes.

And a word to the cleaning before: Vinegar was not such a good idea. Woks are made from reactive metals (that's why they need seasoning) and metals react with acid. Using acid to clean naked reactive metal will result in invisibly small pockets of reaction products on the surface of the wok, mostly rust. You can do this as an early step in cleaning, but afterwards you have to use something to make sure you have removed this too. To peel off the seasoning before reseasoning, you are much better off using a base, as it does not react with the metal. This will ensure that the new seasoning sticks to the metal itself and not to impurities which can be dislodged over time.