Basically the old fan works like this. The third wire on the fan is connected to 2 places on your fan. One spot on the rotor of the fan is electrically connected to Ground. When that place on the rotor of the fan passes one of the 2 spots connected to the pulse wire then it sends a ground out the pulse wire. This will happen twice per revoution.
On your MB there is a circuit that looks for a steady stream of grounds and opens. If it does not see them but sees a constant ground or constant open instead it raise an alarm.
I wish I could tell you that you could take parts off one and put on the other but I do not think you could get that to work. The best I could do is send you a circuit to fool the MB into thinking the fan is still there but then you would never know if the fan quits.
My recommendation is to take the fan back to the store if you can and buy a good 3 wire fan. By the way the way the fan works is a little more complicated than described above but I simplified it because the concept is more important than the actual description.
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