The means of production are mixed between public/state ownership, collective ownership, and private ownership, actually.

I take it that your metric for whether or not a state is socialist is something like, "Worker ownership of the means of production." But that metric has a lot of ambiguities that make it difficult to apply practically in an objective way. Which workers own which means of production, and in what form? Suppose we have a system where everything is state-owned and the state determines who can use what when based on a truly democratic process - but then, an organization of trained professionals in a given field go on strike to demand things be done the way they want. If all the workers should own all the means of production, then the strikers are out of line, but if the workers in a particular field should own the means of production in that field, then the state is out of line.

And should the economy be transformed, fully and immediately, to that ideal? Historically, both the USSR and PRC attempted widespread collectivization of farms, like with the Great Leap Forward, which was an abject failure. That's not to say that farming collectives cannot be successful, but I don't think it's reasonable to expect immediate and total transformation to that model or else a state isn't socialist.

For one thing, virtually every country on earth claims to be democratic, whereas only some claim to be socialist. There are many countries that claim the label of democratic that don't consider the DPRK to be a democracy, but the countries that claim the label of socialist, such as Cuba, generally recognize the DPRK as socialist. If would be strange to refer to a group of countries as socialist and then exclude a country that those countries recognize as being socialist.

It's worth noting that one of the main reasons the DPRK is not considered democratic is not because of the way the government and elections are structured, but because it doesn't allow its elections to be monitored by international observers.