What is the difference between a "political" American President and a "cultural" American President?
This is an incredibly arguable point, and I fully expect my response to be challenged.
Frankly, I've never encountered the phrase "political American President", nor cultural, but the idea of them is clear enough. We should begin by identifying and defining what we mean by "political" and "cultural".
There are a significant number of nuances to these words, but generally speaking, culture refers to the arts, humanities and value systems, while politics refers to the ability to influence others, most typically via public speaking, military force, or economics.
Putting these two words in quotes implies that they do not affect the legitimacy of the President himself; Barack Obama is still the current President regardless of whether he is considered a political, cultural or even a successful one. However, the way that a President conducts his term, and the way that he is perceived and remembered by the public, are broadly open to interpretation.This is where a President might gain these labels.
A "political" President would be one whose administration largely focused on policies, regulations, and other "closed-door" activities that did not typically involve the public, and possibly those who are not directly remembered for their personalities or actions. Examples of these Presidents would include Grant, Taft and especially Coolidge, who is remembered for his silent and dour attitude. The political actions of these Presidents were essential to the history of the United States, but they are not remembered for any remarkable influences on the public.
A "cultural" President would be one whose administration, regardless of its political activities, had a marked effect upon American culture, i.e. "what is American". Obvious candidates would be Lincoln and both Roosevelts, though particularly Theodore, who was well known for his emotional public speeches calling upon Americans to think and act in a certain way. Kennedy and Reagan are also remembered for these traits. While all of these Presidents had significant political elements to their terms (one would hope so), I sincerely doubt that Grant, Taft or Coolidge supersede any of them when it comes to memorability or cultural influence.
It is certainly possible for a President to be both at once, and I think a significant number of them are. Kennedy, for example, was fortunate in that the escalating conflict between the United States and the Soviet Union allowed his political policies to also emphasize our cultural values; we had to question what it was to be American, and how that identity was worth preserving through political action. Contrast this with Woodrow Wilson, who could have been a cultural President but was instead overshadowed by Roosevelt thanks to his policy of inaction; being an American, in the Wilsonian view, meant staying quiet and out of the way of international affairs, a difficult position to defend when your political opponent is an outspoken, world-traveling activist and your constituents are dying in a foreign war.