I think the confusion is that you have two different situations, with two different answers.
Stabilizers don't require a lot of hydraulic power, so operating off a main engine driven pump works fine. It could be either fixed displacement or variable displacement, with the later being more elegant and more expensive. Powering off the main engine removes the need to run the generator all the time, which I think is what you are hoping to accomplish.
The other situation is docking when you want to use your thrusters. Thrusters use significantly more power than stabilizers, and powering them off the main engine is likely not practical. The problem is that when docking your main is at idle, so to get enough flow for the thruster you need a really big pump. Engines and transmissions have limits on how big a pump you can put on them, and that will typically not be enough for good thruster performance. The common solution is to put the pump on an engine that's running at 1800 RPM or more instead of 600 RPM, so that means the generator or a wing engine if you have one. The ideal setup would have the biggest pump that will fit on the main engine to run the stabilizers underway, and to provide partial power to the thrusters while at idle. Then a second pump on the generator (I'll assume no wing engine, but it could go there as well) to provide the additional power needed for full thruster performance.
Now all this really depends on the specifics of your boat. How big is your main engine, how big a pump can it take, what's it's idle RPM, how much power do your thrusters require, and how much more is needed than your main can provide?