Sounds like there are electical problems within your walls causing power to not run as reliably as it should.
The basic problem (the computer starting unexpectedly) has survived the change from one apartment in one village to another apartment in another village, somewhere between four kilometers and three miles apart. The "new" (used) lamp I got for my new desk in my new apartment just made the issue reproducible.
Are they both plugged into the the same power circuit
My "display", a Samsung TV, is not affected by a brown-out. I can switch to TV mode and it runs through without being affected by the lamp, although the lamp is placed exactly between display and computer, with more or less equal distance to both, and the display also is on the same power strip.
does your computer have a remote control to turn it on/off?
The TV has a remote control, but the computer doesn't. The TV would then have to wake the PC via HDMI, but I have not found such a button on the remote. I don't have a blutooth dongle, NFC or WiFi device that I know of. I have disconnected Ethernet but the problem remains.