The half-level points are still there with an ON Semi MC 34063A, but are now a bit different. Now every square starts with at least one pause near the mid level. Sometimes, there are more of those pauses. The difference is that they are usually very, very short, only few tens of nanoseconds in length. There are several types of bad ones, though.
The most common type of square wave is the fast 2 level one. After probing for a while, I'd say that they're more than 90% of captured squares. The is some variation of time it takes to jump from the first voltage level to the second, but it's usually very short.
There are also two more types of square waves I've seen. There's the fast 3 level one. First level dwell time of always very short. There's also a 2 level slow square. It seems to be a bit more common than the 3 level one, but it's still quite rare.
I forgot to show a picture of duty cycle now. It's now mostly around 50%, but goes as low as 40% sometimes and as high as 60% sometimes. At low timebase, the waveform looks healthy.
Also it seems that I forgot to mention the main issue! The large noise peaks are still there. They are not synchronized to any particular point on the MC's cycle. I believe that the noise is just riding on top of the signal MC is creating. The signal on pin 2 of the MC is clean, because I desoldered SMD capacitors I added which connected the MC's power pin to a piece of non soldermasked ground. It seems that somehow the peaks were entering in the MC through them. I added them back for experimental purposes and the peaks on the pin 2 appeared again. Now that I removed them, the peaks on pin 2 are gone too.
The peaks are still the strongest between the TL431 and the "R40" resistor which are almost touching on my PSU. It's also strong near the two 470 µF capacitors marked EC8 and EC5 on my board that are close to 1000 µF electrolytic and L1 coil and in the area near the LM324L. Since this scoping was done on battery power, the area south of the transformer was quiet, as expected.