Why don’t you think the Cruz types are cultural conservatives? If anyone was the champion of cultural conservatism it was Ted Cruz.
When I think of a cultural conservative, I think of someone like Pat Buchanan, Paul Gottfried, Hawley/Cotton (to a lesser extent), and members of the alt right. Ted Cruz is clearly very socially conservative, but there's a difference between being socially conservative and culturally conservative. You can obviously be both. A cultural conservative is also more likely to be a paleoconservative.