I wonder if the types affect the chance of a reaction for some people or in general
Yeah, it does affect the type of severe reaction. AstraZenica had TTS, and with Moderna/Pfizer it's pericarditis/myocarditis.
Chances of it happening are very low for both vaccine types, but when you administer millions of doses, some minority are going to be affected.
They probably mean at the state level which has been consistently led by Republicans since Obama was elected.