At any rate, watch the clip if you haven't recently.
https://youtu.be/n9mzJhvC-8E?t=574
It's pretty interesting actually, to me at least, that Obama roasts Biden pretty much just as well for his off-the-cuff remarks and age immediately after his digs at Trump, which are less derogatory than I had remembered.
But consider the moment right now. Do you think that if President Biden were to satirically mock Jeff Bezos about wealth, space travel, employee care, or putative drone delivery, that he would actually be punching "down" from the white house? I certainly don't think so.
I think that's punching down. I'm not sure it's a helpful abstraction, but I think once one becomes President it's impossible to separate the person from the office until their term ends.
A lot of good that is doing in the Biden years. :/
By contrast, Trump's mocking of, variously, Mexican and central American migrants, the disabled, media reporters etc. is generally seen as "punching down".
In our culture, those further up the ladder are generally expected to take a "punching" (i.e. mockery, jokes, satires, ridicule) from below in good stead. We generally expect them to not "punch down" at those less enabled by power and wealth.