"Slope" implies that the denominator is a distance. In the context of a graph on paper or screen, it is clear how rise and run are mapped to distances. It's a totally sensible definition: slope only makes sense given a choice of axes.
"Rate" implies that the denominator is a timespan. This does not make sense, as there are many slopes which are not rates and which are not presented as rates.
It's kind of a silly thing to worry about, but Khan's answer is unequivocally better.
In math, is not accurate to say the "definition" and the "calculation" are one in the same? Sometimes we require ambiguous non-math language to start understanding a concept but ultimately, once understanding has happened, the calculation and the definition are identical.
Definition and technique to calculate are not the same thing. Consider integer division, which is defined by an equation, but calculated by an incremental long division algorithm.
Most definition use a phrase like "such that", which says nothing at all about calculating.
Khan saying that Karim was wrong, is wrong and actually made him look worse in my eyes. In the video he said "Slope can present rate of change." No, slope is the rate of change. Him saying Karim was wrong about the price vs gigabyte being the inverse if you switch the axes is ridiculous. By definition, you always say the first variable vs the second variable, where the first variable is the y axis and the second is the x. Khan should have left it at that instead of trying to twist things around to make Karim look wrong, it was a poor and transparent attempt at being vindictive.