With my Motorola E (2nd Generation) (also known as Moto E2), the instructions are similar to the other answer to this question:
Open the Moto app:
Disable Moto Display:
Now my phone's screen has stopped turning on whenever I move the phone. I imagine my battery life will increase significantly!
I had the same problem and it was one of the apps I recently installed, possibly a water/drinking reminder App. But as soon as I uninstalled all of my recent apps I downloaded, the problem was solved: no extra pin/unlock screen and no blocking/making incoming calls blurry.