Peculiar noise when observing a square wave
This would not be an answer, but reading through question and comment stream I have a suggestion: reach generator's manufacturer, and report the problem. This way they would be able to know about your issue, prove or disprove it. There're could be other people who may suffer from the same issue, everyone will thank you for reporting. Additionally, you will be able to test manufacturer's support to see and feel if you still want to use their products :)

One more quick thought: coaxial cables and probes may have attenuation switch in them, and you can change their bandwidth. Did you play with it? What I see on your "clearer photo" looks very familiar... kind of impedance mismatch with (rising or falling) edge shooting back into the wire.