A few things aren't right or at least look peculiar. The last images you posted your scope looks like it is in a delayed sweep mode.
From your scope's user manual:
INTEN Although the sweep on the screen is A sweep it indicates B sweep (delay time sweep) by intensity modulation.
The image sort of reflects an intensified portion. Not sure why that is.
Both images should have a much, much cleaner sine wave, similar to the image I posted.
I have never looked at the output of an inverter type generator but like your house mains voltage I can't believe it would be that bad. Since they use an inverter I would expect a little noise but not to that extent.
This should be your scope's user manual.
If you look at pages 27 and 28 they cover modes:
o CHI Only the input signal applied to CHI is displayed.
o CH2, I X-YI Only the input signal applied to CH2 is displayed.
o ALT CHI and CH2 signals are displayed alternately on consecutive sweeps.
o CHOP CHI and CH2 signals are displayed simultaneously by switching between channels at about 250 kHz rate.
o
ADD Displays the algebraic sum of the channel I and channel 2 input signals. If the channel 2 display is inverted (press CH2 display results. CH2 INV), an CHI minus
CH2 INV Inverts the polarity of the channel 2. Useful to comparison of two signals of opposite polarities, and observation of a differential signal of CHI and CH2 along with ADD mode.
Those last few are what we have been getting at. Since the outer shell of your BNC vertical input connectors are at chassis ground using both vertical channels displayed as a single channel can help eliminate a potential shock hazard.
Also page 34 of the linked manual:
CAL O.5V Output terminal of calibration square wave of about 1kHz and O.5V. It has a tip terminal. It is used to calibrate the probe combination .
You may want to try your probes on that since it should be a known signal and see how the display looks. Something isn't right and that needs solved before we worry about looking at signals. Something else to consider is a generator output really only needs to be as good as its intended use or application. I use a few UPS (Uninterruptible Power Supply) which have a less than stellar MSW (Modified Sine Wave) output. Their only purpose is to maintain several electronic devices for a 30 second period until the backup generator is online. Despite the ugly output waveform the supported electronics do fine.
Ron