I have an MSO2072A transformed into MSO2302A (300 MHz). In the same operating configuration, the noise is much more lower than shown by the OP. I'm wandering if his scope is self-calibrated. I had some strange behaviours which have been solved after this self-calibration function (utilities/system menu).
Good thought! I just gave it a try, photo below with the cal in progress. lol - "Last Calibration Time: 2015-9-4". 9/2015 was at least one Rigol software update ago, probably two. But unfortunately no effect on the problem. I put the BNC shorting cap back on (inputs unconnected for calibrate), cranked it up to 500uV/div and 2ns, and the same waveform is there.
BUT.. I did seem to hit a nerve on something. The suggestion was made to try limiting the bandwidth. I set the 20MHz bandwidth limit and the result is the second photo. Looks like normal noise now! The amount and type of noise that Dave describes in his videos (still on high-gain 500uV/div here). Can't get it to trigger either, scanning the trigger voltage level through the whole thing, which may mean random noise (?) as opposed to something periodic. The p-p amplitude appears to be about 1/3 of a division, so maybe 500uV/3 = 166uV. Using the 16uV number calculated for thermal noise earlier in the thread that would be roughly 10x, so probably more going on there than just thermal/Johnson.
Going up to 100MHz with the bandwdith limit makes the amplitude about 30% larger, but still appears to be mostly random noise with just the occasional flash of a defined waveform. Then removing the BW limit entirely and I'm back to that original large(r), complex, waveform.
So a few of quick theories come to mind.

One is EMI entry at the higher frequencies, as has been suggested, but keep in mind a BNC shorting cap is on the input. The EMI would have to be getting into the internals of the scope, past any shielding. At such a high level of gain though it is probably quite possible. Another idea is something to do with the 200MHz software hack. Keep in mind this is a 100MHz scope, DS2102, software hacked to 200MHz, That might explain why it seems to be more-or-less behaving up to 100MHz. Yet another idea is that PLL and PLL instability described in the Yaigol thread affecting the ADC. This is the "older" scope hardware, the non-A (DS2102 not DS2102A). It may be more likely to have that PLL problem. I'm just assuming fixing the PLL may be one thing Rigol did with the change to the "A" version, more stuff than just adding the 50R input option. That might explain why yours doesn't appear to have the noise waveform, being the "A" version. This one is running the latest software though, or at least what was latest last I checked, 00.03.05, HW = 2.0, from the system info screen.