I feel I should jump in and rectify one thing here though.
In most major game companies, how do the people that design game concepts (how much recoil will this gun have? What should we name this city? Should we add X gameplay element? etc.) get those positions?
The initial question is this, and not, how to become a game designer. The one reason I'm saying this is because, in many cases (if not most), the Game Designer doesn't actually decide a lot of this stuff.
I've been at many places, and have had many answers for this question, but overall:
The designer might choose initially how much recoil this gun will have, but an executive committee will probably have a stage gate to express their comments in one way or another about the game, asking for vague or very specific changes. In the case of client-oriented development (vendor), the client might outright ask for changes, perhaps even on a daily basis. So, while, in essence, the game designer might effectively make such a decision, the sheer amount of changes demanded will mean the game designer doesn't really own the vision. While he can be the ideas guy (though not only game designers have ideas), he won't have the final say in anything.
In the case of a lore-heavy game, if its an established brand, some kind of brand manager or licensor will probably show up and demand that everything goes through him for approval. This is usually the case for marginally big brands/companies. It would definitely be the case if you work on an Elder Scroll or Castlevania title for example.
If its a new brand, you'll probably have a game director of some kind that will be present. He might have come up with the original concept, or may rely on a team to come up with the answers, but he will probably have the final say.
So, in essence, the closest roles that could fit your demand are (with descriptions based on my experience only, so they may very well vary!)
- Game Director - 10+ years of experience and/or outstanding all-around proven skills. The Game director is in charge of the vision, and shouldn't be considered just the 'boss of design'. He is a force that 'opposes' the producer in that he seeks quality and has 'unreasonable' demands. The core idea is to have the producer and him sit and talk things through to always make sure the priorities are up-to-date and that the game will be as good as it can be within budget and time constraints. Sometimes called Product Owner, though many stakeholders can be called that, so I prefer to go for something a bit more specific.
- Executive(s) - Someone not directly linked to production, possibly a CEO, CTO, Vice-President (of any department). Someone with an eye for the product, but from the outside. He has a stake in the company, and his words are heavy. Without his approval, the project won't move along. Possibly the longest route to become the ideas guy, especially since you won't be tossing many ideas, mostly just 'destroying' the ones that are laid in front of your if they suck. From personal experience though, while their criticism is much welcomed, their actual ideas tend to suck.
- Brand Manager - Especially in large company hierarchies, for established brands. Will require a very in-depth understanding of the brand, possibly because you've worked on it for several years. Could be as specific as being the 'commander keen brand manager', but generally encompasses larger brands. Is also generally associated with strategic thinking. For example 'is it a good year to launch a new Assassin's Creed title?' It's important to note that marketing will also play a role in his day-to-day and that sometimes the 'when' (the game is released) and 'how' (its hyped) will play a more important role than the actual 'what' (the game is about).
- Licensor - I've seen this more rarely, but in the case of an intellectual property that is owned by a 3rd party that has given rights of exploitation to a developer, they generally retain control over everything and may have you send everything over for approval. Your best chance of becoming a licensor however is to come up with a great idea, build it yourself, sell it out, and then start discussing with big companies once you've made millions and seek to extend the brand without actually lifting a finger
Of course, game designers are idea-guys in their own right, but so are programmers, artists, QAs, etc. The best game designers I know aren't really ideas guys, they're more ideas channelers. They take what fits best, know when to say no, make sure everything they do answers an objectives and prevent feature creep. Then, they document everything in such a way that all scenarios are accounted for, well explained, and make sure everyone has the same understanding of the concept. They do wireframes and flowcharts, create (theoretical) balancing documents, etc. I feel they're much more about putting ideas to life than actually being ideas guys...