for those new to the industry you must also understand that there are levels/steps within the career of a game designers and this can create confusion when describing what a game designer is.
On AAA larger projects at times (rough generalization) the more junior you are the more detail work you do… the higher up you are as a designer the more high-level concept-idea vision-holding decision-making your role is
junior designer, game designer, senior designer, lead designer, (at times design manager), creative director, (for multi studio companies or at least mutli project big company there is also a Chief Creative Director)
Within the general Game Designer figure, yes of course you have a number of parallel/complementary spcializations especially on large projects (UI designer, level designer, system designer (variations), specific mechanics desigers, etc…). Also the game designer is not always the story writer… often professional story/script writers are contracted to work that part under suprevision or in collaboration with the design team
if you have decision power or partial decision power on which games gets made varies from Company to Company…
like in other disciplines, the higher your position the bigger the managerial side of your job as well
the higher your position as a designer the closer you are to a Movie Director as mentioned in another post… also different companies have slightly different Job Names… for example in Company A the game director is the creative director… in Company B they may be two different people
at times it is hard to understand or accept that the Idea Man does exist, indeed inside and outside the industry most developers assume they are or can be good game designers… just as probably most people assume they can be good managers or whatever…
P.S. update: about the ‘dreaded Idea Man’… do not confuse how the term is used on the ‘Collaboration Forum’ and the value of such figure for a Commercial Studio.
When someone new goes to the collaboration forum and says…
“I have a great idea, I need a team to make this game… etc…”
You do get general replies like:
“So you are the Idea Man? What else can you do?”
this is because you are within an indie environment where most are trying to make their own games often for both passion and financial return. and most people do have plenty of game ideas.
And to top it up, the new guy with the idea is most times just that, a new guy with no ideaMan-history behind him 
For a Commercial Game Studio the situation is different, they want to employ the right person for the job (in each discipline). if they need a good game designer or visionary with good grasp of what works what doesn’t and why… they look for the best candidate to fill the gap of what is often called the idea man.
Oh by the way… it is surely a full time job, often a 24/7 job in some case and it does come with its own share of responsibilities, merits, stress, iterative processes, constant evolution, and what not…