At this point the problem seem independent by the power supply if with battery is worst...
No.
I recommend you take some time now and really read all information about this issue and filtrer also out all this "information" what only mix and shake this case.Also it is good to read some amount theory for understand these things so that all is not so much just "magic" things.
If "lemon" show this last image so that mains power is disconnected and scope run using battery, situation and noise is different.
With battery AND mains power together produce more noise and this noise component what rise is main PSU flyback circuit noise.
I repeat it agen here and agen with some different words.:
Inside scope there is several noise sources. And all noises is conducted to different parts with radiation and conductice, common mode and differential mode. And for understand full case it need understand about RF fields and how RF trawels in different cases and circuits.
Noise "sources":
main PSU
8.4V main PSU flyback converter (1). (severe noise source)
(nominal 8.4V but around over 6V up to to 8.4 if battery) This is master power line and from this is all others derived.
If mains power line is out, battery give this voltage
(1) this is off if only power source is battery.
sub SMPS circuits: (voltages not exactly)
-7.6V (severe)
Adapter board
+5V (moderate - severe)
+3.3V (it seems that this is moderate or less)
TFT back light SMPS (severe)
All these generate high frequency noise in these units what can name real "noisy GND" problem units.
Every these noises have some amount different signature.
All these can separately detected and analysed as example I have done.
These highest spikes in "lemon" last image come exactly from main PSU running flyback converter what produce 8.4V from example 230V Ac or what ever is connected to mains power line.
(these spikes also vary dependent of battery charging sitiuation but they are always more high if battery is installed and powered by mains. (in problem units)
If we look externally oscilloscope whole GND it have sum of these noises and it all is (what are meaningful noises) RF noise from some MHz to over 200MHz. And this we can look that it is "common mode noise". Simplify... it can think that scope GND area works like you have transmitter and it is its antenna. Now you connect wire to it... there start flow RF current to this wire and also it is now part of this "antenna" network. it radiate (meand current flow) and it conduct to places where it is connected... and current flow... etc
But all wires have inductance. You know what happend when RF try go via inductance.... it produce voltage over it...
Oscilloscope probe have also wire.. this coaxial from BNC to probe tip. This RF noise travel via this probe cable and then it travel via this probe separate GND wire and it continue its travel forward to where you have connected it. Important is that in this point where scope probe GND wire is separated (this blacl 15cm wire) there is now This RF travel over this GND wire and RF produce voltage over this GND wire. Yes... this GND wire is short copper wire. How its ends have differet voltage... it is nearly zero ohm. For 100 or 200MHz it not at all "zero ohm..." if it is thinn or thick,, no big change.. if it is gold or iron.. not big difference.. you can coat it by silver.. no big difference (in this case). But if you make it more short... it change radically its inductance and so also reactance. Ferrite over power cord.. yes it is useful but not for this what this probe do. And you can take 1000 euro brand name high-end passive proble and it is more noisy in this case..
Yes your DVM show zero ohm over this short GND wire. But... this wire is inductor...it is coil (without turns) every wire is .. 1cm or 10m have inductance. Independent of if it is cheap Owon wire or expensive Rolls-Royce wire or Tektronix wire.
This RF flow over this wire and it produce voltage over it! And what our oscilloscope show, it show voltage diffrencies between probe tip and probe GND. If you think keep tip in zero potential tightly and then you run signal to GND wire... oops what you see... you see signal.
If think ferrites on power cord or some other place it need understand how they work and how this noise itself.
Primary solution is reduce this noise there where it is produced. (repair these SMPS circuits what are guilty for this)
Secondary solution is try reduce its affect to measurement = do not externally connect this signal to signal under test. And here we can do it with short GND and also get some more help using ferrites over probe coaxial. (2-3 turn thru ferrite example. It may block some amount of this RF noise flow from scope GND via probe cable to probe tip end.
perhaps "lemon" can show how this second picture how it change if take power cord out and run with battery alone.
This proofs that with battery assembled and used with mains power situation is more bad tah mains alone without battery os with battery alone without mains power.