VideoHi Martin,
you asked about your common ground problems with the two way rectifier and maybe these comments can make you understand a little bit what went actually wrong.
@15:00 you may have first measured the AC-Voltage between GND of the oscilloscope and GND of the function generator. Due to induction on the main supply cables, there can actually be some voltage, which could even cause more problems (see:
Wikipedia, "Ground Loop").
Up to here, you have explained the basic problem pretty nice, but later in the video it seems that you do not have listened at all, to what you have said

@18:20: "this line over here is at the same potential as this"
You have not checked the AC voltage (same potential), but if there is not much ground loop currency, you may try to connect the crocodile clip of the oscilloscope probe to the black banana GND connector from the function generator. This could reduce the noise. You could also twist the green and black cables between the coax and the bread board ("twisted pair"), but due to the small distance, it should not really make much difference.
@20:44: Here starts your confusion. The wave form displayed by the oscilloscope is exactly what it is supposed be, it is not what you would
like to see.
I have named the points in your circuit with letters:
Diagram-1:

The yellow sinus wave is the voltage between A and G [A|G]. What you would like to see as second signal (measured by the UNI-T) is the voltage from P(lus) to M(inus) [P|M]. But due to the common ground problem of the oscilloscope, you can only get the display of the voltage between P and G. Therefor the green curve is actually a combination of the sinus wave and the
current-voltage characteristic of the diode [P|G].
If you would like to see the signal between P and M, you may connect one channel to P and the other to M. The difference ([P|G] - [M|G]) will be the camel's hump, that you are searching for. Maybe you can save the original sinus [A|G] in memory and display this together with that difference (P-M). To be honest, i never worked with a digital oscilloscope.
@21:22 Bumm! Here is the point, where you create a half wave short circuit. This is exactly what Dave Jones warned you about. Only the fact, that your function generator is not a powerful source, prevents you from blowing up your system.
Diagram-2:

Connecting G to M creates a short circuit over the diode between M and G [M=G]. Now, the second half wave flows as current right through the diode from M to A and the voltage [A|M] drops to the forward voltage of this diode.
@21:45: "now there are two earth points being referenced"
Autsch! That's your problem. Because you connect them at different potentials into your circuit!
@21:49: "the signal generator is confusing things"
No, the generator is innocent, because it is half wave shorted out. You are the one who is confused

There is only one earth. It seems to me, that you did not take Dave's advice seriously, maybe because here is not the kind of power involved, that could blowup your GND cables. If your generator would have more power, you would have at least toasted the diode [A|M]. Just because there is no smoke does not mean, that every thing is fine. The short circuit is real and it destroys half of your wave form and thus the frequency you expected.
@22:00: I do not really understand what you mean by the inverter, but your problem here can not simply be solved by isolating the function generator from the oscilloscope. The real reason behind the problem is, that you want to display the two signals [A|G)] and [P|M] simultaneously. This is why you would need differential probes. Your oscilloscope does not have two independent pairs of cables (A|G and P|M), but only (Channel1, Channel2 and one GND). There is only one earth.
The transformer complicates the situation here a little bit. It introduces a small phase shift, which obviously effects just the second half of the wave.
@23:49 "the sinus wave has got a lot of noise on it"
No, please do NOT call this noise. Noise is random (a statistic phenomena), but this is obviously a part (distortion) of the wave form, which is a result of the transformer.
@25:xx "[...] and i'm taking again the signal [with the UNI-T] at the same point as my channel 1 [...]"
Here, you got me really confused
It took me a while to understand that you just lied;). What you sayed is NOT correct. While the UNI-T is sitting between A and G, Cannel 1 (the yellow curve on the scope) is obviously grounded at point M (and not G)! So it is not the input of the rectifier, but rather the voltage between A and M. This is just the voltage [A|M] along the left upper diode (the light red one in the diagrams). It is the sinus signal during the first half wave (when the diode resists), and the forward voltage of [A|M] for the second half of the wave (where the diode is letting the current pass though).
Diagram-3:

@25:19 "i haven't connected the crocodile earth clip ..."
Yes you have!
The GND of the channel 2's probe is already connected to point M! Unless you spend the money for a second earth (differential probes:)) the common ground is already connected. Remember: the crocodile of channel 2 bites as hard as the crocodile of channel 1
@25:24 "i clip that [second GND of the oscilloscope at a different place] on now, and i show you the result"
Bumm! that's the classic look of another half wave short circuit! You have again fallen into the same trap that you described at the beginning of the video.
@25:52 " frequency now is kind'a hooked up to be the same", because you eliminated the second half of the wave with your short circuit. The second half wave of the green curve is flat as a dead mans hart beat

@26:36 Now you seem to have disconnected the oscilloscopes GND from M. The expected result is seen in diagram-4:
a) the yellow signal [A|G] rises to the full input wave (including the distortion of the transformer).
b) the green signal shows the voltage at the diode [P|G], which is:
- the sinus signal (minus the forward voltage of diode [A|P] during the first half of the wave, and
- the forward voltage of diode [P|G] for the second half (when current flows from G to P).
Diagram-4:

You may visit Dave's video again and keep in mind, that it does not take a dramatic effect like blowing up cables, to cause harm to a circuit. Sometimes just the partial clipping of a wave form, can damage your thought process and lead you to false assumptions.
EEVblog #279 - How NOT To Blow Up Your Oscilloscope! I hope, this could answer some of your questions and clear the smoke of the unburned cables out of your mind

Regards,
Stephan
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Edit: Embedded attached images.