I agree with this. I am not American either, but your President is like the #1 spokesperson for your country.
For the American's saying he doesn't define the US, it is like looking in the mirror. You may think something looks great when you look in the mirror, then you ask all the people that are close to you (mom, dad, best friend) and they all think you look awesome. However, an objective outsider can clearly see that your colors clash, your ass looks fat, and your shirt is 3 sizes too big.
I think when we're thinking simply politically, of course a country is defined by the head of state, but broader, when thinking about that country we're more likely to think of the cultural contributions.
When I think of what defines a country like Italy, I don't think of Paolo Gentiloni the Prime Minister (and truthfully, I had to Google who the PM was), I think of wine, food, and fashion. When thinking of Australia, I think of the country as being defined by beer and beaches, not Malcolm Turnbull. The UK, I think of pork pie, double-decker busses, pubs, and phone boxes, not Theresa May (although in fairness, I do think of the Queen as well, though I think for semi-permanent heads of state or dictators [not that the Queen is a dictator] this is probably more likely). Likewise, I think when somebody wants to stick it to a country they say, "Well, yeah, of course your unpopular leader defines your country that I want to dislike," but in actuality when they're not trying to be argumentative, they're more likely to think of those persistent brands or features that persist far longer than the temporary head of state. For billions of people around the world, the Nike swoosh or McDonalds arches are more recognizably American than whoever the president is.
Though, this phenomenon of associating cultural contributions to a country rather than its sitting head of state could also be different for people looking in at the US, rather than me, looking out at other countries from the US. US politics generally gets far more media coverage in Italy than Italian politics gets in the United States, and so our top politicians might become household names after an election.