Your definition of Socialism is pretty narrow. The traditional definition basically just boiled down to "a system where the workers are in charge of their production", which encompasses a lot more ideologies and economic systems than just "everything is owned by everyone", Marxist style.

On the other hand, you're applying a much broader definition of "means of production" than is typically discussed in Socialist circles. Most Socialists don't have any significant issue with personal possessions, or the presumption of exclusive use of things that you consistently are using. The bigger concern for Socialists is rent-seeking on said right of exclusive use.

Basically, only a small subset of Socialists would have any issue with "that's Bob's hammer, don't use it without asking". Most Socialists would be opposed to "that's Bob's hammer, you can't use it unless you toss him a fiver".

An Anarchist commune is approximately peak Socialism, everything else is an (arguably) necessary compromise with the complexities of large scale society.