As a millennial, I define socialism as the funding of increased government-run social services through a progressive taxation system.
That's not socialism. That's a welfare state.
If you want to ramp it up to social democracy, add in some nationalization of what's called the commanding heights. Jeremy Corbyn's Labour Party is embarking upon re-nationalizing such things.
If you want to ramp it up to socialism on the national level, nationalize practically everything. However, socialism can mean simply public ownership and control, including at the most local level, like a small village or even a totally rural area. In that case, it's communal ownership typically by an entity that's recognized as a legitimate public government.
There are variations on the theme, but there are limits as to what constitutes a proper definition of socialism.
Some people would also include ownership by the whole public of a given area even if that owning entity is not recognized as a legal government. I don't have a problem with that so long as everyone in the region or area has equal ownership and control: democratic. In that case, the owned thing is socialistically owned.
A co-op that isn't universally owned in the manner described wouldn't constitute a true socialistic entity under my definition. To me, defining such an entity as socialism is to water down the meaning to a harmful extent for achieving socialism: equal democratic ownership and control as the only form in the entire state, including nation-state and even supranational-state.
Lastly, central planning is definitely not required and dictatorship is absolutely forbidden, as it runs contrary to the whole people controlling (democracy). That's why Marx was flat out wrong. Violent revolution and the dictatorship of the proletariat were both really bad ideas. Killing and then excluding survivors was a bad idea. Neither was a means for winning them over, just the opposite.