I like this question; it shows that the primary difference between when to use definite and indefinite articles isn't always as cut-and-dried as standard rules might indicate. As is said here often, English is a flexible language, and most rules have plenty of exceptions that apply in certain contexts.
In this case, could the writer have used the instead? Sure. However, if that were my sentence, and I used a definite article, I would punctuate it differently:
On Tuesday, voters re-elected the president, who promised to fight for higher taxes on the wealthy, for more public investment and for careful cuts in spending.
Notice the added comma after the word president.
I like the original wording better, though; it seems to flow better. Although the indefinite article makes the sentence appear to have been written in a general sense, it really hasn't.
For example, if I say:
Voters re-elected a president who promised to fight for higher taxes on the wealthy, for more public investment and for careful cuts in spending.
then that sentence, taken by itself, could indeed refer to any president, of any country, at any time in history, who got re-elected on that platform. It could be talking about, say, former president Bill Clinton, or to Sauli Niinistö of Finland.
What makes it refer to Barack Obama, then, instead of some other president? In a word, context. In this case, that context is provided by a few key facts:
- This sentence was published in a newspaper, not a history book
- The newspaper is a U.S. publication
- The editorial was published just days after the U.S. 2012 election
- The sentence begins with, "On Tuesday..."
Taken together, those four pieces of information make it very obvious that the phrase “a president” refers to Barack Obama, the 2012 incumbent U.S. president. So, the sentence works. Even though the indefinite article would allow that sentence to refer to any president, that particular sentence doesn't refer to any president – context won't permit that.
As a footnote, this is why I've consistently praised the way you write questions on ELU. Had you merely said, "In this sentence, who does the article refer to?" then the question might have stirred a contentious debate. Instead, you began with a citation:
The New York Times article (November 9) titled, “The Fiscal Cliff Opener”...
Some might wonder, "Why would I need to exclude extraneous information like that?" No – that information is not extraneous, it's vital to understanding the sentence. Otherwise, the exact same sentence, word-for-word, could refer to something entirely different:
On Tuesday, voters re-elected a president who promised to fight for higher taxes on the wealthy, for more public investment and for careful cuts in spending. On Wednesday, riots had erupted in the streets of Dankoon, leaving dozens of people dead, and over 100 buildings burned to the ground. President LeMonde wasn't going to let that happen this time, which is why, on the eve of Election Day, soldiers had been deployed to every major city across the province. Post-election rioting wasn't going to become a tradition in Denov – not under his watch.