These all are indeed just few examples of different mechanisms on how to get the disturbance there. The common part of all of these is, it is the cable shielding, which is supposed to carry all these currents and so leave it out from the signal wires. And in that way make sure, the residual current in the signal wires will remain safe for the interface IC's.

So you may create some shields or separations inside of the lantern, so the lantern may stop to interfere with the monitor. But then there would be some other spike in the network, which will couple into the signal wires and destroy the interface IC's.
That is, why I have called that lantern "a messenger" and not a cause: It's disturbance just gives you a message, than your HDMI connection is really very bad and needs to be fixed...

I am not questioning that the cable is the main fault here - that is clear in the first place

Qestion is, is this a sign of the "messenger" having a problem as well that should be fixed ? such as the open PE and isolation breakdown which are present in all the cases above

According to your pictures, the problem would be mainly in the house installation - open PE.
For the lantern that does not mean anything, the intrinsic capacitances are there so, the coupling does have the path as depicted available even without any other fault.

Would the current going to that path be significant with ok PE ?

There will be some current, but nothing harmful. If the PE would go all of it's length along the corresponding L and N through the "fuse box" till the point where they are connected together (usually at the connection to the distribution cable), the voltage would be minimum. However the voltage range the digital interface may tolerate (to work with) is just +/-2V including very fast transients, without good GND connection in the cable that will be exceeded really easily.

By the way you have depicted the current path just from the connection between the ballast and lamp, but way higher capacitance is between the arc and ground (the same capacitance responsible for an easy lamp ignition), just because of the rather large lamp surface diameter. You should not forget the arc starts to build up from the hot end, so just before ignition the complete tube becomes one large conductor connected to the hot end, being discharged once the plasma reaches the cold end...


But still these currents are rather low compare to e.g. real hard short circuit fault currents (kA range). And these should be handled by the mains wiring plus signal cables as well (probably the signal disturbance could be tolerated in that case, but it should still not damage the interface; and that limit is in the 5V range, so no more than 5x the operating range; so if 5kA in the PE should not damage it, 1kA won't even disturb the signal, while the lamp won't generate more than 1A peak)