Composite video from rear car camera - bad signal (wrong levels?)
I've solved the problem by supplying the camera lower supply voltage (8.5V) by using DC-DC step down converter. It seems that signal generated by camera is better or more acceptable for display to recognize as proper video signal. @Justme also provided explanation why that could be the the case:

The video signal peak amplitude and average depend on how bright areas there are in the picture. That and also the used supply voltage can have an effect on the overall DC bias of the video signal, and that's why sometimes the display can and cannot lock on to the signal.