A registered party member is just someone who voted in the primary.

The Democratic primaries in my area had a bunch of uncontested seats or I was comfortable with all the candidates, so I registered Republican to vote against Trump.

I'm a registered Republican, but I am not a supporter of the GOP.

Consider this. As this was his first presidential election to vote in, registering Democrat meant voting in a meaningless primary, so registering Republican would be a strategic choice to vote against Trump as a nominee.

Is this the case? We don't know yet, but voter registration for a 20 year old doesn't infer much

So I have two (conflicting) thoughts regarding people as right wing.

Firstly, here in the States, right-wing encompasses a lot of spectrum, from the extreme reactionary / accelerationist folks that are keen to replicate the worst of the atrocities of the German Reich, to those who are neoliberal / neoconservative, and believes in a multi-faceted economy, a pluralist, secular culture and the use of military adventurism to secure American interests abroad when diplomacy and trade relations fail to bring those about.

Crooks could come from anywhere in this gamut, whether he is (what I suspect) an extremist glad to serve the new American order as a militant or an operative on the field, or someone who is Republican because they identify as [R] but still wish to see the US as a place where democracy and government by the people prevails. (It isn't, with our democratic features tightly constrained, but the rival party prevents one party from taking absolute power.)

Secondly, when it comes to Republican intellegentsia, there's a tendency to want their party to be less extreme than they have become, as we've seen with those Republicans like Brad Raffensburger who is not willing to cheat or collaborate with Republican officials to bend the system to secure an election victory, but who still believes in the anti-poor, anti-labor, pro-prison, pro-police Republican policies that laid the rails to our current situation on the precipice of autocratic takeover. We have a lot of people who knowingly voted in Reagan, George H. W. Bush, George W. Bush and Donald J. Trump, and kept voting Republicans down-ballot, aware that the policies they preferred would eventually drive the US to the kind of general precarity and discontent that leads to systemic breakdown and autocratic rule.

But that's not where Crooks is, as much as we like to imagine our operatives are playing 5D chess, whether they're allies or enemies. Essentially, Crooks did as most terrorists and rampage killers did, and committed angry suicide, with as little actual planning as he could manage.

But that could change if Mr. Wrench chatted with him on line for twenty hours over the course of four weeks convincing him to take a pot-shot at Trump, or if he's done a lot of reading of political theory and came to such a conclusion on his own. But until that evidence comes up, I think he was just acting out of pure rage based on personal circumstances.