I have not been able to get a straight answer on this one.
One definition would be that an electrical designer is a particular type of electrical engineer. An electrical designer is generally someone who designs electrical systems, which is one branch of electrical engineering, but there are many others.
For example I could call myself an electrical engineer (in the sense that I am entitled to place certain relevant letters after my name) but I could not describe myself as an electrical designer because that's not the job I do.
A lot of times a designer is a technician type of position requiring a two year degree. An electrical designer could specialize in many different areas including robotics, PLCs, computers, etc. Designers usually apply theory. Electrical engineers learn a good deal of theory in the pursuit of their 4 (or more) yr degree.
The work and title depend a lot upon the company you work for.
The site I've referenced below is specifically for K-12 but I recommend it to a lot of people looking for clarity between Engineers and Engineering Technologists and the sort.