My thought is that the minister we saw in Demons/Terra Prime was some kind of diplomat, and not the member of a parliament.
Well, there's no reason he can't be both. Charles Flanagan is serving concurrently as both the Minister for Foreign Affairs and Trade of Ireland and as the Teachta Dála for Laois. Stéphane Dion is both the Canadian Minister of Foreign Affairs and the Member of Parliament for Saint-Laurent.
It seems unlikely to me that Nathan Samuels (the character you are thinking of) was a member of the United Earth civil service, though. The title of "Minister" in governmental systems that use it is generally reserved for members of the Cabinet who head the various executive departments (such as a foreign ministry) -- and in parliamentary systems, those people are usually concurrently serving as Members of Parliament. So it seem likely to me that Nathan Samuels was a Member of the United Earth Parliament, in addition to whatever other office he was holding.
On top of that -- he seemed to be directing United Earth's foreign policy in "Demons/Terra Prime" rather than merely executing the policy a higher-up had devised. So to me, that implies that he's not just a civil servant; he's exercising the powers of state.
The novels actually do refer to him as Earth's Prime Minister. Which would make some amount of sense, it would explain why he had the authority he did over Archer, including the ability to assign another captain to the NX-01.
I mean -- sort-of? The novels establish Samuels as United Earth Prime Minister, but they are deliberately vague as to whether or not he was Prime Minister during "Demons/Terra Prime." Canonically, he was only ever addressed as "Minister." And as I noted above, he did seem to be directing United Earth's foreign policy rather than merely executing someone else's policies.
However -- it's notable that Hoshi was legally empowered to nullify his orders aboard the NX-01 bridge when he tried to issue orders to the crew. As Hoshi noted, Smauels was not part of the United Earth Starfleet chain of command.
To me, that implies two possibilities:
1) That Samuels was the United Earth Foreign Minister during "Demons/Terra Prime" but not the Prime Minister; and/or,
2) That the United Earth Prime Minister is not legally part of the UESF chain of command.
Neither possibility precludes the other. In many parliamentary republics, the Prime Minister isn't part of the military chain of command; rather, the President (or sometimes the Defense Minister) serves as commander-in-chief. The President or Defense Minister will issue the orders the Prime Minister wants them to (and sometimes Prime Ministers will even appoint themselves as concurrent Defense Ministers, as Winston Churchill did when he served concurrently as Prime Minister and Defense Minister during World War II), but legally the P.M. isn't actually part of the chain of command.
Personally, I just assume that Samuels was United Earth Foreign Minister during "Demons/Terra Prime" and that his party won a general election held right afterwards, becoming Prime Minister just in time for the books. But that's just me.