2 is fine.

I know where you’re coming from, I use a closed notebook with external display sometimes.

I might need to be more specific. If the notebook is used as a typical notebook, and one closes it with the intent of putting it to sleep, once sleep has been reached an external mouse should not wake it.

However if you do toggle the power settings to allow the machine to function with the lid closed and/or machine docked then you do want to mouse to wake it keep awake the machine.