That is more like the definition of "communism".  

To me, socialism is more general term that contrasts to "capitalism", in the sense of bein more society-oriented rather than individual-oriented.  Among other things:

(1) each individual should be rewarded by society in proportion to what he does for society, rather than by his possestions, descent, titles, intelligence, shrewdness, etc.;  

(2) property and economical rights of the individual are not absolute but are subordinate to the interests of society as a whole,

(3) the state is supposed to provide public services like health care, education, social security, transportation infrastructure, emergency and security services, etc.;

(4) the state should try to ensure equal opportunities to everybody and ensure that everybody has a decent minimal living conditions.

Socialism definitely does not imply state ownership of the means of production; but it implies state regulation, e.g. to force companies who provide vital services like water or electricity to charge reasonable prices, respect quality standards, provide basic service even to unprofitable areas, etc..  It admits, but does not require, that such services be provided directly by the state, by civil servants or through state-owned companies.

Socialism implies protection of consumer rights and mandatory product quality and safety standards; but is quite compatible with free market economy.  In fact, as part of protecting consumer rights, socialism implies state intervention when needed to keep markets free, by preventing the formation of monopolies and cartels.