Yeah, that’s definitely a usage of the word socialism I have heard, but it is not generally a definition most socialists or socialist parties would use and it has some issues in my opinion.
This is such a broad definition of socialism as to make it almost meaningless, as this definition fits every nation on earth today and most through history. ancient Rome used public money to fund public roads, subsidized grain for the poor, public entertainment and land grants for veterans, public aqueducts, and other public programs, yet this was 2000 years before the concept of socialism was really invented and I don’t think anyone is holding up ancient Rome as an example of a socialist society.
I would define socialism by two characteristics. One is control over the political economy by the proletariat (workers), as opposed to the bourgeoisie (capitalists/financiers/business owners). In a bourgeois run capitalist state, there is still publicly funded services, but they generally set up to benefit privately run industry (public highways, government subsidized research, police) or they are concessions won by the proletariat through class struggle (universal healthcare, social welfare programs).
The second characteristic is economic organization around common need, rather than around the pursuit of profits. This would require taking the means off production (factories, businesses, utilities, etc) out of the hands of the bourgeoisie.