Following the advice of 2 reddit users, I've tested out using a cable built out of seperate smaller joints, instead of 1 single joint.
It was actually more stable than I thought, but the weight of the cable had too much effect on the physics in my opinion.
That's why I have decided to use the best of both ways. Technically the cable is now working as a single joint, but this joint is working invisibly. On top of this I have simulated multiple connected joints that visualize the cable hanging, while not actually affecting the physics.

Here's the result of my work! I'm really happy with the cable movements. 
You can see the single physical joint as yellow line when the debug mode is on.

I won't post for a while after this. My daughter will be born soon, so I'm taking a break for a couple of weeks.