Lack of square wave here is not an issue, it never is square on analog parts of an oscillator circuit, should actually be more sinusoidal. The chip will make square out of it internally. The second image is already half-way there, that's probably the output, could be also affected by the scope and probe limitations, as well as your choice of grounding point for the probe. That being said these resonators are more noisy in their output than crystal ones. But this is good enough, if a bit low on the amplitude, but that might just be how it works - hard to tell.
The frequency is a problem though. Quick calculation and 820 (I got 833 from counting the divisions, close enough) by 492 is 1,6(6). This multiplied by 300 gives 500, which is pretty much what you found testing the rpm via INDEX signal. So the resonator is running way too fast, at least according to the datasheet for this chip. I suppose though if this drive used a different configuration of the motor coils it might need a different frequency. I think I see a number "819" on the resonator, wonder if that has anything to do with the frequency. Usually these parts have unusual markings that do not clearly state the frequency so I'm not convinced. This might be a problem, or just so happens that we ended up with these numbers.
I need to refresh my own memory on these resonator specs, because different servo chips can require different frequencies. Some newer 3.5" drives even use a clock signal provided from the main board. In the meantime can you try to figure out the color bands on C2 and C4? These look like green resistors but should be stable ceramic capacitors (in somewhat unusual packages). Or make a close-up photo. The light reflections make it hard for me to read the values, but these sure aren't 47pF, which would indicate the values were also adapted to the different resonator frequency.
EDIT: Actually a frequency of 819200Hz would make sense for this chip as well, considering the dividers it is using. 819200/(8*1024)=100. Which then works nice with x5 and x6 multipliers to get the rotation speed. It just needs a different motor than what the datasheet wants. And the signal level is more or less in the range too, input is higher than 2mVpp.
If you are up to some more scope measurements, probe pins 1, 2 and 3 of the servo chip. Might tell us more about that suspect capacitor.