Socialism (defined as no private ownership of 'means of production')
Thank you for clarifying where your mistake was. Socialism does not require there be no private ownership of means of production. Socialism does not seek to make everything state owned, rather it seeks to ensure that opportunity for the individual is not hampered by their inability to obtain capital. In fact the most significant European examples of socialism are also countries with significant privately-owned industries - think of Volvo and Saab from Sweden, Nokia from Finland, the list goes on. The government never controlled those companies.
Communism, on the other hand, does seek to take control of the means of production. The two are not interchangeable.
at a national level requires a command economy
That statement only makes sense given your false statement about socialism that you made earlier. A command economy is absolutely not required for socialism.
No price signals, no profit motive
Again, look at the products of actual socialist countries and you will see how wrong you are with that statement.
Command economies all have excessive concentration of power. It is just a fact.
While the states that called themselves "communists" ended up that way, it is not the result of Communism itself. Rather it is because in a large state, you will almost without fail end up with someone rising up in the power vacuum created by the pursuit of Communism who will attempt to concentrate power. At that point the "communist" state collapses into Fascism, only without the name. This is no different from the fate that the US would face under many of the "libertarian" dream states that have been sold to us on TV and youtube in recent years.