I used to use this definition, but it has a few key issues. Modes of Production should be defined in a manner that is consistent. If we hold this definition for Socialism, then either it means a portion of the economy can be Socialist, ie USPS, or a worker cooperative, or it means an economy is only Socialist if all property has been collectivized.
For the former, this definition fails to take into account the context to which portions of the economy play in the broader scope, and therefore which class holds the power in society. A worker cooperative in the US, ultimately, must deal with Capitalist elements of the economy. Whether it be from the raw materials they use being from non-cooperatives, to the distributors they deal with, to the banks where they gain the seed Capital, they exist as a cog in a broader system dominated by Capitalists in the US. Same with USPS, which exists in a country where heavy industry and resources are privatized, it serves as a way to subsidize transport for Capitalists. The overall power in a system must be judged.
For the latter, this "one drop" rule, if equally applied, means Feudalism and Capitalism have never existed either. There is no reason Socialism should be judged any differently from Capitalism or Feudalism.
What Socialism ultimately is is a system where the Working Class is in control, and public ownership is the principle aspect of society. If a rubber ball factory is privately owned but the rubber factory is public, the public sector holds more power over the economy. In the Nordics, heavy industry is privatized for the most part, and social safety nets are funded through loans and ownership of industry in the Global South, similar to being a landlord in country form. In the PRC, heavy industry and large industry is squarely in the hands of the public, which is why Capitalists are subservient to the State, rather than the other way around.
As for the purpose of Socialism, it is improving the lives of the working class in material and measurable ways. Public ownership is a tool, one especially effective at higher degrees of development. Markets and private ownership are a tool, one that can be utilized more effectively at lower stages in development. Like fire, private ownership presents real danger in giving Capitalists more power, but also like fire this does not mean we cannot harness it and should avoid it entirely, provided the proper precautions are taken.
Moreover, markets are destined to centralize. Markets erase their own foundations. The reason public ownership is a goal for Marxists is because of this centralizing factor, as industry gets more complex public ownership increasingly becomes more efficient and effective. Just because you can publicly own something doesn't mean the act of ownership improves metrics like life expectancy and literacy, public ownership isn't some holy experience that gives workers magic powers. Public ownership and Private ownership are tools that play a role in society, and we believe Public Ownership is undeniably the way to go at higher phases in development because it becomes necessary, not because it has mystical properties.
Ultimately, it boils down to mindsets of dogmatism or pragmatism. Concepts like "true Socialism" treat Marx as a religious prophet, while going against Marx's analysis! This is why studying Historical and Dialectical Materialism is important, as it explains the why of Marxism and Socialism in a manner that can be used for real development of the Working Class and real liberation. When taken consistently, AES states do in fact fit into the categorization of "Socialist," even your original definition would categorize them as such.