UX designer
Decides how a person gets from their question to the action: what sits where, how many steps checkout has, what happens on error. This is not the same as graphic design and the two are constantly confused.
When your project needs this person
On a shop, on a more complex service, on any site where people must choose between several options.
What happens when they are missing
A beautiful site nobody buys from. Checkout has seven steps, the important information is at the bottom, and the contact form asks for six fields, two of which are pointless. Each one sheds part of the audience.
How this appears in an offer
Appears as "prototype", "wireframe" or "user journey". If the offer says only "design", you are probably discussing looks, not structure.
What they do NOT do
Does not pick the colours or draw the logo. That is graphic design, a different profession.