From what I can tell the post seams to want to know how you handle yourself more then how you design games. Though they are quite closely related. The position of game designer is more then likely the most sought after job in the industry and also requires the most talent. You need to know how to do everything. In general you need at least these skills:
You need to plan everything and keep track of everything. The GDD and TDD will create your game in text. Everything is documented from hours worked each day to the x,y location of the menu buttons on the main menu yet alone the file structure, naming conventions and assigning who does what. This is most likely the hardest part of the job
It's one thing to be able to assign everything and a completely different beast in keeping track of everything. You are in charge of knowing where everything is at any given time when it's coming in and where it is stored once it is in. Also you need to know what build you are on and when the next update will be.
You must be able to explain what you want, why you want it and when you want it. Your team can not read your mind so you must show them what you mean. Videogames tend to be very visual so we draw a lot of pictures and graphical representations in our documents.
Also you must know the lingo of all the divisions (a general summary would be art, modeling, programming, testing) so that you can effectively know what is going on and suggest ways to get around bugs.
When things go wrong, and they always do, your in charge of fixing it. You need backup plans for everything and then backup plans for those backup plans.
Here is another place where knowing all the fields works great. Say the engine you are using only supports 96dpi images for some reason. The programmers will know that there is a draw error along the lines of "file structure not supported" Thats clear to the programmer but if you tell the artist "your file structure is not supported" you will more then likely get something like "well can't you program it to be supported"
But because you are a awesome Lead your know what the error means and you can tell your artist "Your file structure is incorrect, please make sure when you save it out that the specifications match that on the GDD"
Your also in charge of scheduling due dates for assets and builds. This all needs to be documented and planned so that everything moves along smoothly