I'm virtually positive that the top video in post #4 is the fuel pump. I've been dealing with one on a 328. Fortunately it's not my car, and they had already made the decision to replace the fuel pump before I got involved. The designs are not that dissimilar. I replaced the fuel pump, replaced fuses, replaced at the fuel injector relay (last two being cheap), and I believe I finally have the problem identified.
The way I understand it, and it appears to be showing in your video, is that when you click the remote to unlock the car, (turn the key to accessory in older models) a fuel pump electronic module called the EPK sends a signal to the fuel pump which causes it to run for a few seconds in order to have pressured fuel in the fuel injectors for start-up.
In every video that has talked about the fuel pump, part of the diagnosis (basic mechanic not code use obviously) of the fuel pump is to listen for it to make that sound. It's absence says that the pump isn't working. On all the models that I've seen, the fuel pump is under the passenger side rear seat so to that end, when it's a model like yours, you stand by the passenger's side rear door when you unlock the car to listen for the sound. It's quiet evident in your quiet garage.
I have no idea what the sound is in the second video. It could be an evolving fuel pump issue or it might be something else that is activated at the same time.
I would have no reason to believe that the sound in the first video leads to the 2nd.
Last edited by attycda; 12-15-2019 at 04:38 PM.