Your characterization is even more misleading than theirs. Do you seriously think Trump created the vaccine himself and it didn't go through the FDA and scientists or anything else?
The mere existence of the vaccine should be enough for her but no if Trump said something she won't take it.
This is an English subtlety. There is an implied negation of the other condition and an implied independence.
“If my wife tells me to eat cauliflower I will. If you tell me to, I won’t.”
A literal interpretation is that if you tell me to eat cauliflower I will avoid cauliflower altogether. The natural English interpretation translates to “Your endorsement of cauliflower will mean nothing to my choice of action”. Funny, eh?
The mere existence of the vaccine should be enough for her but no if Trump said something she won't take it.