That signal looks OK. I don't think the signal is "dirty", but it seems you didn't set a proper trigger (at least in the first screenshot it clearly looks this way). The trigger setting indicator at the top right corner says "rising edge, channel 1, level 0V", which means the oscilloscope synchronizes with the point in time the signal on channel 1 crosses 0V on a rising edge, i.e. goes positive from being negative. The signal never is negative, though, so the oscilloscope doesn't trigger on the signal at all. As it is in the "AUTO" mode, it still "auto"matically captures some waveforms to help you find out why it doesn't trigger. If it were in "NORM"al mode, it would not display anything at all. You should put the trigger level to ~2.5V for this signal to be displayed cleanly - well, unless you use "AC coupling" for the trigger source, but I get the impression you didn't do that.
Anyway, as the OSC frequency seems to be correct (quantized to a period of 70.0ns instead of the expected value of 69.8ns), the OSC signal is likely not the problem. Furthermore, as your EGA card works correctly in a different system, the fault is likely not on the card. Can you show a photo of the "wrong geometry"? I understood this as "the aspect ratio is wrong", but that wouldn't prevent you from reading the time in DOS. At the moment, I ponder the idea that possibly some data bit for I/O port writes is not properly sent over the ISA bus, mis-initializing the card. OTOH, if the ISA bus is that broken, the 5152 shouldn't be able to boot as well.