Your interpretation requires re-defining the word "president" to mean something it simply does not mean.
That's a pretty strange thing to say, considering that "president" is a very commonly used and indeed abused job description in a variety of organizations, the field politics only representing a tiny corner of the general field of applications.
The ill-defined, wishy-washy structure of the Federation would be a natural habitat for a president or three, in the sense of "president of the society", "president of the board", "president of the council". No real need to assume that a president in this context would be a head of state or anything. The President in ST4:TVH did seem to do rather menial things for a head of state, after all - presiding over a court, concerning himself with the affairs of a single planet, etc.
We can of course decide that the two onscreen Presidents, from ST4 and DS9, were sitting on one and the same chair, with broad powers. Or we can say the UFP has half a dozen Presidents of various sorts, and the one declaring martial law isn't necessarily the one presiding over courts of law or diplomatic rows or whatnot. Or we can say the role of the UFP President changed in between. But there's no inherent merit in insisting that the role should fit a pattern established by the United Earth or the United States, since neither of those organizations holds the patent to the concept of president.