From an electrical point of view what matters is adequate copper to carry the current and adequate insulation to prevent short-circuits and electric shock. From that point of view any adequately rated cable could be used provided that they are connected through correctly.
The problem is that insulation colours generally have an implied meaning. e.g. Brown = live, blue = neutral and green/yellow for earth. If you were to extend the pump motor wiring with your white cable you or someone else may, in the future, assume that the green/yellow is an earth whereas it is live. This is a good enough reason not to use it.
I recommend you find a three-core + earth cable. Connect the earth to the motor chassis and the earth terminal in the controller. The other three should maintain the correct relationship to the original wiring.
Figure 1. Capacitor start induction motor wiring. Source: DIY.SE.
The diagram of Figure 1 may help you understand what you are actually wiring. The motor has two windings - one powered from L and N and the other wired L - capacitor - N. The diagram also shows how to change the direction of rotation - which you don't want! The diagram does not include the earth connection which you should include.
If you're going to do some non-standard wiring then label both ends clearly and permanently.
DIY cable labels. Source: How to make DIY bread-clips to organise your cables and power cords.