Using an admin account as the sender for an action rule instead of site:current-user
I was able to select the wanted sender User by typing in the USER ID when the form is in direct input mode. To get to this "direct input mode", use the button "switch to direct input mode", as shown here:



After doing so, the form will look like so:



So in the end, the result of doing so is that if you read your private messages, then the sender is not the new user anymore, but the user you selected by ID manually (as shown in the above form).