The US legal code regarding threatening a president is 18 U.S. Code § 871 - Threats against President and successors to the Presidency. The first clause defines how threatening the president is illegal. The second clause starts off by defining "President-elect":
The terms “President-elect” and “Vice President-elect” as used in this section shall mean such persons as are the apparent successful candidates for the offices of President and Vice President, respectively, as ascertained from the results of the general elections held to determine the electors of President and Vice President in accordance with title 3, United States Code, sections 1 and 2.
So the key here is that they are the "the apparent successful candidates […] as ascertained from the results of the general elections." It doesn't define this phrase though, especially the word "apparent".
It appears that the prosecutors in the case mentioned in the question are arguing that Donald Trump became the apparent President-elect when, after Election Day, voting results began coming in and he was projected to win 270 electoral votes. Given the amount of stock that most Americans seem to put into the media's coverage of who wins the election (and the fact that the person in question seems to have assumed he would be president), that's unlikely to be contested.
