Although it's true that a designer will not necessarily be a "master" at any one thing, trying to get your typical programmer types to relate to your typical art types can often require a bridge.
Can we say a game designer is a mediator between code(rs) and art then?
To a degree. That's not fully accurate though. A designer can and should fill that role, but it's really a designer's job to be the one who does the creative work - world building, designing gameplay systems and features, coming up with concepts for the story and settings, characters, etc. Ideally those are then passed onto more specialized people, like writers and programmers. More than anything it is a problem-solving job, with the central problem being "how do we make a fun game?" and then solving all the sub-problems in their millions that appear along the way. As Machocruz said, this can and should involve some degree of technical skill, though the extensiveness of this will very dramatically.
Depending on the size of the company and the type of game being made, a designer will do much more than just the simple creative brainstorming. For instance, designers are often responsible for scripting gameplay, doing level design and creation, defining raw data which is then fed to the game engine (think spreadsheets defining various statistics), doing the majority of writing for the game and even sometimes music and sound. The smaller the company, the more general a designer will have to be in skill set, while at large companies like Ubisoft and EA you tend to get people whose job is basically just telling other people "do that, that would be cool." I've read interviews with some industry veterans that are pretty enlightening - one where (and forgive me for forgetting his name) the guy did everything from art, to sound, to writing, to scripting, to some programming, to level design, etc. on an early-90s RPG, and when he told his interviewers at a large game developer, they were literally stunned that he did all that himself.
That's yet another reason for having so many misconceptions about what game design is and what game designers actually do - it's extremely variable based on the individual job.
On a personal level, I can do
level design and building, some scripting, I can write and communicate rather well (= game design docs and coordinating team members), I can use Photoshop and similar programs, have some basic knowledge of 3D modeling programs, and so on. I don't know how that stacks up to other designers but it's definitely served me well in building various game mods etc. over the years. I imagine that skill set is probably wider than what you get at some larger studios, though. The problem for a designer is that you are only as useful as your skill set allows for a certain project... if "designer" means "gameplay programmer" and you aren't a programmer, then you're going to be out of luck, or will have to be a very fast learner.
