from wikipedia:
>a form of coup d'état in which a political leader, having come to power through legal means, stays in power through illegal means through the actions of themselves and/or their supporters.
So Trump trying to stay on in 2020 was an attempt at one. Trump delegating firing people to Musk isn't really. The current stuff doesn't seem illegal either - I think the US president is allowed to say find out who's unnecessary and fire them even if the people being fired shout 'coup', 'unfair' and the like. I'm not sure it's a good idea but that doesn't mean it's illegal.
Congress has Constitutional vetting power over Presidential appointments. It's part of the checks&balances. That has not happened. The purpose of a department has been altered by an appointee who has not been vetted. Again, incredibly problematic if not outright illegal.