Why can't Safari display this emoji when I am logged in?
The problem is that auto activation of fonts had been disabled. FontExplorer X Pro was the culprit. 

Safari seems to need auto activation to be able to display emoji correctly.

The solution is to manually re-enable auto activation. In Terminal:

atsutil autoactivation -e

Credit: https://discussions.apple.com/message/32553150#message32553150