OK, I have some more information, as well as a bunch of photos of traces 6-16 measured in both the working and faulty scopes.
1) I exchanged the timing boards A4 between the two scopes, but it did not cure the faulty scope, so the problem is not there.
2) I have measured traces 6-16 on each scope by using the other as the measurement device. The faulty scope can still be used as a measurement device as long as one is aware of the problem. Only flat traces will be limited in horizontal extent because then it is not triggering. Whenever there is a nontrivial waveform it will trigger and behave as normally. Note that the timebase calibration might be a bit different between the two, especially since there is something wrong with the "CAL" knob for the timebase on the working scope. But I believe it is minimal. The CAL knob on the working scope does not click into place, but I believe it is in the "calibrated state" even if the knob looks a bit randomly rotated in the photos (it usually has no effect on the trace).
I have grounded the input on the scope that is analysed, because I have a problem even with no signal on the faulty scope. The scope that is measured is always set to 0.1ms on the timebase. The measurement scope is set so that there is 1V/DIV vertically except in the photo of trace 10 because that is a big signal. The timebase is 0.2 ms/DIV on the measurement scope unless otherwise stated. It can also be seen in the photos. Also note that I have adjusted the vertical position so that 0V is two divisions below the horizontal centreline (because then I get most of the traces in DC mode without having to adjust vertically or change the Volts/DIV)
First, here is the signal when not probing anything, so you can see what is the 0V reference:
Working scope:

Faulty scope:

Comment: When using the faulty scope as a measurement device here, the trace is short because the waveform is flat and then there is no triggering. This is the problem I'm trying to solve.
Trace 6:
Working scope:

Faulty scope:

Ignoring the horizontal problem when using the faulty scope to measure, these are the same. Around the same postive DC voltage, around 3.3 VDC.
Trace 7:
Working scope:

Faulty scope:

Comment: Here there is a clear difference: the low and high voltages are about the same (3.4V and 4.3V), but the faulty scope is doing something strange. The HI state on the working scope lasts 0.2ms, but on the faulty scope it is much shorter, and occurs more often. The latter is something that happens in several of the traces below. In this photo, I wonder if I forgot to get it correctly triggered, and perhaps this is freerunning. I will check again next time I open the scope.
Trace 8:
Working scope:

Faulty scope:

Comment: Around the same low and high voltages (0V and a bit more than 3V), but longer lasting high voltages on the working scope. Higher frequencies of short high voltages on the faulty scope.
Here is a time zoom with 1 uS/DV of the short pulse happening in the faulty scope:

Trace 9:
Working scope:

Faulty scope:

Comment: As in trace 8, more frequent, and much shorter pulses on the faulty scope. Voltages are about the same.
Here is a time zoom to 5 us/DIV to show the structure of the pulse happening in the faulty scope:

Trace 10:
Working scope:

Faulty scope:

Comment: 5V/DIV vertically since it is a big signal. The top voltage is the same, around 12V. The faulty scope doesn't start from the bottom of the voltage ramp.
Trace 11:
Working scope:

Faulty scope:

Comment: The same low and high voltages, but the faulty scope jumps down to often, and doesn't stay down there at the low voltage.
Time zoom at 50 us/DIV on the short pulse happening in the faulty scope:

Trace 12:
Working scope:

Faulty scope:

Comment: The same here. Too frequent and the low voltage period is absent. In the service manual, the low voltage period here is the HOLDOFF period, so it is completely missing on the faulty scope.
Time zoom to 50 uS/DIV showing the short pulse on the faulty scope:

It looks just like in the service manual, except it is missing the HOLDOFF period.
Trace 13:
Working scope:

Faulty scope:

Comment: The faulty scope has just a flat line here...
Trace 14:
Working scope:

Faulty scope:

Comment: Both show a flat line. The horizontal problem appears when measureing with the faulty scope because it is freerunning due to the flat signal.
Trace 15:
Working scope:

Faulty scope:

Comment: As other traces above. The lov voltage periode is shortened on the faulty scope, and occurs more often.
(Attempt at) time zoom on the short pulse happening on the faulty scope:

Trace 16:
Working scope:

Faulty scope:

Comment: Both are flat at around the same voltage.
My main observations are as follows:
1) Waveform voltages seem fine.
2) Trace 8 shows that the faulty scope doesn't have a correct "A retrace" period. Or, it is very short and happening too frequently.
3) Trace 10 seems very much correlated to chat is happening on the screen. The voltage ramp is shortened, as well as the trace width.
4) Trace 11 shows that the faulty scope has no "A end of sweep" period of low voltage.
5) Trace 12 shows that the faulty scope has no "HOLDOFF" period, but the waveform up until that is supposed to happen looks fine (apart from duration, of course).
6) The voltage ramp-up that should happen in trace 13 is missing. This missing voltage ramp-up is called "HOLDOFF TIMING" in the service manual.
7) Trace 15 should have a period of low voltage, but it is drastically shorteden and happens too often, similarly to several of the other traces.
So, does this info narrow down the list of suspected components?

As mentioned above, trying the known healthy timing board A4 from the working scope did not cure the problem, so the problem is likely not here.
Best regards,
Torquil Sørensen