Lets get a naming scheme so it sounds less like "colour by numbers"
The motor controller (MC) has 4 wires:
MCbk (black) which sounds like the ground connection
MCy (Yellow) which sounds like start/stop
MCr (Red) which sounds like a speed control
MCbe (Blue) which id EDC and I'm not sure what that is (does it measure distance, or perhaps cause it to stop after a certain distance?)
Then on your Golf cart controls (GC) you have
GCr (red) to the on/off switch
GCb (blue) to the potentometer
Whilst it seems that MCy should connect to GCr, and MCr should connect to GCb, a connection to MCbk seems to be required.
Is the metal of the golf cart itself the common connection between the GC controls?
If you connect a multimeter (on an ohms range) between GCr and the cart, do you see a connection formed when you press the start button?
Similarly, do you see a variable resistance between the cart and GCb when you do whatever is required to change the speed setting?
Thirdly (and this may be independent of the others above) If you take a reading between the CGr and CGb cables, do you see a connection of any sort which only exists when the start button is on, and which changes resistance as you move the speed control?
If the first two are as I expect (so should the last one, but it's not significant) then you probably need to connect MCbk to the metal of the cart -- but be careful in case another part of the controller is grounded (although this seems unlikely since this point is labelled gnd according to your description).
As long as MCbe can be left disconnected, the cart should now do *something*.
However there is another potential problem. The potentiometer controlling the speed on your golf cart may not have the value that is required for the new controller. It is possible that all of the speed control will happen over a very short range of the speed controllers movement, or that it goes flat out with little control over speed. If either of these occur, then the pot may need to be changed to a value which suits the motor controller.
I presume your golf cart is powered from a small lead acid battery. These can deliver an amazing amount of current, and the risk is that if the controller is wired incorrectly, that current may be allowed to flow somewhere it should not, and damage may result.
Be very sure that you're happy that the connections are correct before you attach the battery. If the golf cart has a fuse for the battery, it may be a good idea to replace that with a much smaller one for your "smoke test" (when you first attach the battery) so that (hopefully a incorrect wiring will low the fuse before it destroys anything else).