A lot depend on who is doing the asking. These job titles are flexible, and there s a certain snot factor in having a fancy title.

If it were me -
Web designer = graphics/visuals almost exclusively.
Developer = overall / high level design and prototype storyboarding. May or may not include coding.
Programmer = writes code.

But don't go by that. Even I would not.

In most places, you will do it all. And needs vary. One person may do everything for a small site. A huge site may have even more specialties.

Just see if the job description and salary fits. Never mind the title. Old joke:

Boss: "Jenkins, we've decided to give you a higher title instead of a raise."