Earlier quoted context omitted.
But by the all-encompassing definition you have here, having a military is socialism. Paying bus fare is socialism. Running elections is socialism. I'd argue that such a definition of socialism is so expansive as to be worthless.
By the classical definition, a government-paid military and city-run buses are absolutely socialist organizations. Running elections is not a means of production, so it can't be socialist.
I agree totally.
As for the military, I'm not so sure. As far as I know every military is run by a state. A private army is generally called milita. A military is also not really providing value to the people, but rather providing value to the government, sometimes over the people. Mandatory public military on the other hand might qualify for that term.