OK, a couple of things. First, the video did help, but there are two misunderstandings about scope theory that are apparent.
First, I didn't elaborate, but when you switch to the GND, you are disconnecting the inputs internally and shunting the preamplifier to ground. Once you verify that the inputs are unresponsive, you know that that feature is OK and there's no point in supplying any inputs.
Second, in XY mode, those 4 pushbuttons don't do anything. They don't change the channel selection, for example, which I saw you pushing.
You can and should look at the schematics that I referenced and look at the diagrams of at least the switches to see if you can understand how that works.
In any case, we've determined that the problem seems to be on the input side. I also noticed the strange behavior as you were changing the CH2/X attentuator level. It's not uncommon to see a brief jump in the dot as you change ranges, but on CH2/X, it breaks up or jumps at a 45-degree angle--but that channel should only affect horizontal position. So there's some leakage or crossover between channels and it looks roughly equal.
We can also try some testing in the normal, non-XY mode. If you leave the inputs grounded, all 4 pushbutton out and turn XY off, what happens?
Do you have a 9-volt battery, or preferably, several of them? I'd like to try using a larger input signal through the attenuators to try and track down the area where the noise is generated.