I have friends from Hong Kong who always use steel woks in their take-away.
A cast-iron wok would require a different technique for stir frying. It would be slower to heat up and retain heat when you didn't want the food to continue cooking.
Stick with the one you have ...
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.
Best Answer
I can't speak authoritatively, but I do have all three types and have had good luck with my seasonings, so I'll share what I do.
For cast iron, I use solid vegetable fat exclusively (Crisco). I did the original seasoning by coating it in fat and baking in the oven. To clean it, I use salt, Crisco and a paper towel to get any food bits off. I then get the pan hot and wipe it down with a little more Crisco. It's not PTFE nonstick, but pretty good.
For my forged iron DeBuyer fry pan, I followed the manufacturer's recommendation for initial seasoning: put enough oil in the pan (I use canola) to coat the bottom. Heat to smoking. I swirled the hot oil around to coat the sides a bit. Let it cool, pour and wipe out the oil.
The thing I've found is it takes quite a bit of use to develop the seasoning. Here's a picture after about a month's use - note how dark the sides are getting (the bottom is less dark because I learned a green scrubby pad (Scotch-Brite) will remove the seasoning entirely. Oops!).
I did some scrambled eggs for the sake of science, without adding any extra oil to the pan. They behaved as nicely as any PTFE coated pan I've ever used.
Today the entire bottom of the pan is that rich mahogany color, and a fried egg slides around in it like one of those goofy AS SEEN ON TV ads.
EDIT: Used the pan for over easy eggs this morning and snapped a new picture. This is about 3 months after the last photo.
Normally I can just wipe it out with a paper towel. If I've been cooking bacon or something that left residue, I'll run water into the pan while it's still ripping hot, then wipe it out gently with a sponge.
After that I heat the pan up and add just a tiny bit of canola oil. I wipe the oil around with a paper towel and put the pan away.
For carbon steel--like my wok--I treat it exactly the same as the DeBuyer pan. Clean gently, after each use get the pan ripping hot and wipe it down with some oil.