Maybe I can add something.
In a previous life I was an Automation Engineer with a systems integrator for any type of manufacturing machinery from nacho cheese packaging to rocket propellant lab testing to you name it. I was responsible for estimation, hardware procurement, electrical design, logic programming, occasional panel building, occasional installation, FATs and SATs, checkout, startup, and emergency troubleshooting.
Now I'm an Automation Engineer for a petroleum service company and am restricted to programming a single system, on a single platform. I do not touch any hardware, do not troubleshoot any devices, and do not have to soothe any client headaches.
So to point, in my experience the title Automation Engineer can have a vastly different meaning to different organizations.