If the external display wakes up as your main display with the same desktop picture and icons as your laptop it means the internal display is asleep.
If your mouse is laggy and display looks fuzzy it means your external is mirroring your internal display and that your internal display is on. Turn mirroring off if that happens.
If you’re worried about heat (lid shut mode blocks the vent that runs the width of the keyboard) you can leave your lid open and use your external display normally.
I have not noticed a heat problem using my 2013 MBP Retina in lid shut mode after my two year old threw a phone at the screen (kids! Lol)
If your mouse is laggy and display looks fuzzy it means your external is mirroring your internal display and that your internal display is on. Turn mirroring off if that happens.
If you’re worried about heat (lid shut mode blocks the vent that runs the width of the keyboard) you can leave your lid open and use your external display normally.
I have not noticed a heat problem using my 2013 MBP Retina in lid shut mode after my two year old threw a phone at the screen (kids! Lol)