My intuition is that:
1) "I'm president of the United States!" would be chosen to put more focus or emphasis on the power invested in the role - and less on the person. Read: How could anyone question the power of the person (me in this case) holding this office. I am shocked!
2) "I'm the president of the United States" might be chosen to emphasize the singularity of the office holder. Read: There is no one other than me who can make this decision, since I have that power exclusively.
While either intuition may be wrong, the point is that you would have to look at how the sentence is used in the context of the conversation its a part of. Grammar choices are constrained by the context - not so much by simple 'style'.
At the risk if being repetitive; the decision is a result of the speaker's intent in that context.
Now, whether the constraints of grammar might restrict what or how can think or speak. Well...that's a debate for another day.