Configure X properly to start GUI programs from SSH session

I've installed xorg, xdm and fluxbox onto a GUI-less minimal Debian virtual machine instance (on GCP), then I've tried to run an application chromium or google-chrome with GUI. The execution of the program fails with the following message:

$ google-chrome
[2631:2631:0905/040834.818761:ERROR:ozone_platform_x11.cc(240)] Missing X server or $DISPLAY
[2631:2631:0905/040834.821056:ERROR:env.cc(255)] The platform failed to initialize.  Exiting.

If I connect via RDP and open a terminal on GUI then enter the same command, the browser application simply starts as expexted. If I enter this command from a **putty session**, the program starts in the **RDP session**.

$ DISPLAY=:10.0 google-chrome

I want to be able to launch programs what use GUI from SSH. I understand that I need to configure a display related variable, but I don't understand exactly where and how. - should I hardcode this display variable into .xinitrc or .xsession script? - if so, this wouldn't destroy the dynamic display handling behavior of the xorg? - or is there a more advanced approach how should this work properly? - isn't there any tutorial (for dummies from scratch) about how this whole xserver should be configured properly? --- I've copied my .xinitrc and .xsession files from the internet (because the installation was not smooth), which is probably is not good for the above requirement: My .xinitrc file is:

#!/bin/bash
if [ -d /etc/X11/xinit/xinitrc.d ]; then
  for f in /etc/X11/xinit/xinitrc.d/*; do
    [ -x "$f" ] && . "$f"
  done
  unset f
fi
exec fluxbox

My .xsession file is:

exec startfluxbox