That what you are seeing is probably a common mode transient caused by you switching of the function gen with the power switch. You will probably see the same signal even with the output disabled.
It is too slow to be a common mode transient. Those are typically in sub-microsecond time spans. What you see in the screenshots is typical for less than optimal power-supply sequencing at power down.
Except that nobody can reproduce it in such form.
And you that don't own that AWG (or even similar one) are making conclusions based on some scope images that are made God know how. Scope was sampling at 2Ms/s using Peak detect mode. Any high frequency would be sampled as outside envelope, not as base frequency..
I don't have SDG1000X either. But believe RF-Loops results, who made VERY comprehensive tests at the time. He knows it inside out.
If some specific test could be defined he could run it.
But the way I see it, whatever is on the scope screen has more to do with what and how was tested.
You and I might disagree from time to time but one thing I agree with you 100% is your signature.
OP should connect scope to an AWG directly, use full memory, maximum sample rate, and try then.
To exclude DUT, it's power supply, connection to computer ground via ICP connector etc etc..
When something is wrong, you reduce variables. Eliminate sources of confusion. Reduce problem complexity.