How do you call someone that analyses and optimizes websites?

You’re describing a consultant. But I would say spend the money on a good web design team that can do “branding.”

Yes, I feel the consulting term should be used here. Perhaps “web optimization consultant”, a bit long but I think it works fine here. Anyone reading this on your resume is likely to continue to read your summary of job responsibilities as that title, and it will become clear.

Well, the thing is not that I do this. We are looking for people that do this, or a course that can be followed about this subject. A creative team can create a nice website, but then it still needs to be optimized for marketing reasons.

I think dynamicduo nailed it.

I just asked my daughter a similar question last night. Her husband does something like what you described above for the University of Florida. He creates programs and updates them. His official job title is Software Expert.

Found it:
User Experience Expert