In TVH it's the Federation President that issues the planetary distress call. It's possible that there is also an Eath President who sent a different distress call but there's usually one official spokesperson for the government.
Yeah, but the United Earth government is a separate government from the Federation government. That's why the Federation is a
federation rather than a unitary state.
For a roughly analogous situation, when Hurricane Katrina hit in 2005, both the Governor of the State of Louisiana and the President of the United States issued separate declarations of states of emergency.
And, of course, issuing a distress signal -- which in this case was primarily intended to warn incoming ships away from Earth -- is
not the same thing as legally declaring an emergency.
Uhura and Spock did mention multiple distress calls but they could be from private ships.
But one could also have come from the UE government. There is, again, no reason to presume the United Earth presidency doesn't exist.
To add another real-life example, here in Italy we have Presidents for
everything.
We have the President of the Republic (ceremonial head of state), the President of the Council of Ministers (government leader - prime minister), the President of the Chamber of Deputies (speaker of the lower house), the President of the Senate (chairman of the upper house), the Region Presidents (governors of regional units), the Province Presidents (governors of sub-regional units), the President of the Supreme Council of the Judiciary (self-governing body of the judiciary), the President of the Constitutional Court, even the senior Judge of a court is called the President. Fuck, in Italy even
boroughs have a President.

Pretty much any position, private or public, with "Chief", "Chair", "Main", "Prime" or "First" in the name translates as
Presidente in Italian.
Heh. Yeah, and here in the United States, we have more Presidents than we tend to think. There's the President of the United States, there's the President of the United States Senate, there's the President Pro Tempore of the United States Senate, the Presidents of the Parishes of the State of Louisiana (Louisiana's equivalent to counties), Presidents of various villages in the State of Illinois, and Presidents of the Boroughs of the City of New York.
So it's not weird for me to think of a President of Earth subordinate to the President of the UFP.
Well, you know, "subordinate" may not be the best phrase. In a federation, the provincial/regional/state governments aren't
exactly subordinate to the federation government. They have areas of exclusive jurisdiction and share sovereignty with the central government.
But, yeah, point taken -- there's no reason there can't be a mostly-ceremonial-with-reserve-powers President of United Earth alongside a Prime Minister of United Earth within the Federation. Just like there's no reason that there can't be a President of Deneva and President of Trill.