Is it ok to use a metal turner in a wok

Induction cooking works by generating an electric current in the metal cooking vessel and converting that current into heat, which requires a resistive material (i.e. a poor conductor).

It's a bit of a catch-22, because you need a good conductor to actually distribute that heat. This is why some of the best induction cookware is clad metal - two layers of (magnetic) steel around an inner layer of highly-conductive aluminum or sometimes copper, sometimes layered multiple times this way. The thick magnetic sheets generate heat, and the thin conductive sheets transfer it.

Thicker metal means that the surface is slightly farther away from the magnetic field source, but also has significantly more resistance (since resistance is proportional to length, and we are talking about doubling or even tripling that), so theoretically it should be much more effective at converting the electric current to heat, and heat up faster.

The disadvantage of a thicker metal - assuming that there is no additional heat transfer material - is hot spots. This doesn't matter at all for a skillet, but if you're trying to cook with, say, a cast-iron stock pot or dutch oven, you'll find that the hot spots are even worse on an induction cooktop than an electric or gas cooktop, since the surface heat is generated very rapidly but takes forever to spread.

Carbon steel is a pretty good compromise, which is why it's generally the material of choice for woks, and that's the material I'd choose for an induction cooktop for any vessel other than a skillet or frying pan (assuming I had to choose a single material; again, copper/aluminum-clad stainless steel works better). For pans/skillets, you want the heat to stay focused on the surface, which makes heavy/thick cast iron a better choice.

I can't honestly say that I've had much experience with forged iron, but all my instincts are telling me not to bother with it because it would be combining the worst of both worlds - lower resistance at the surface but still relatively high resistance up the sides. The only caveat to this would be a possibly significant difference between the magnetic susceptibility/permeability of forged vs. cast iron, but this is likely going to vary from vessel to vessel anyway (not all cast/forged iron is the same) so I think your best bet there is to try sticking a magnet to it. If it sticks much harder to the forged than cast iron (doubtful) then it might be good for surface cooking; I still wouldn't choose it for a larger pot.

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.