Is slightly different, you have a party vote/appointment for the candidate for prime minister, from elected party ranks, which is similar to the US presidency, apart from the president not having to be elected to government already.

But then the presidential vote doesn't vote directly for the president, but elects electors, who then elect the president, which seems an unneccessary step given you already have the vote results from each state and only has any effect at all if an elector does not vote the way the have been campaining.

I am not saying it is any more or less democratic than my government, I am just saying that this detail seems weird.