> By definition, a conservative is a slow-motion progressive
No, they aren’t. A conservative is an anti-progressive. (Classically, an anti-liberal, but that’s less clear these days, because while classical liberalism was a progressive position for its time of origin, “liberal” has been overloaded to refer to defense of the status quo elites established by the success of classical liberalism and continued progress in the direction laid out by classical liberalism, with the latter use sometimes more specifically narrowed to be “at a slower pace than is described by ‘progressivism’”.) The former sense overlaps with conservatism, the latter sense with (possibly specifically slow-motion) progressivism.
> Ignoring that technicality, there are 3 types of republican:
I would say more than that:
1) social conservative (often, but not always, for religious reasons; this group was basically drawn in as a side effect of targeting group #5 with the southern strategy)
2) libertarian (as a distinct faction, really drawn as strategists realized that it was easy to adapt the decentralization and States Rights rhetoric that was classic dogwhistles for group #5 along with a lot of the anti-business-regulatory rhetoric designed to give mass appeal to the interests of group #3 to appeal to this group and reinforce the Republican coalition)
3) corporate neoliberal capitalist (the oldest major faction of the Republican Party, it has been a major factor since very close to the founding of the party, and was dominant from about the time the slavery issue was settled until the 1960s Southern Strategy)
4) American nationalist (there’s actually a lot of these in both major parties; some late cold war issues, though, gave Reps an advantage here.)
5) White supremacist/nationalist (historically a strong Democratic consistuency from the founding of the Democratic-Republican Party with occasional rifts that healed because they had nowhere else to go (like the 1940s Dixiecrat split) until Johnson signed on to the Civil Rights Act and then the Republicans targeted this group heavily in the Southern Strategy (preventing that from being another temporary rift.)