You should do
mf <- model.frame(price ~ carat + cut, data = diamonds)
y <- mf$price
x <- model.matrix(price ~ carat + cut, data = mf)
or at least that is basically what the rstanarm and brms R packages do when fitting models like this. If you want to get rid of those Helmert contrasts, you can first coerce cut to an unordered factor or call
options(contrasts = c(unordered = 'contr.treatment',
ordered = 'contr.treatment'))