Remember social democrat friends, keeping capitalism as an engine of growth means keeping the engine of global warming and keeping imperialism. But then again, it's merely dark skinned people who overwhelmingly suffer from those...

It emerged to placate the working class in the West at the expense of "Third World" peoples. It succeeded in getting western workers to support capitalism and the intensified imperialism that funded social-democratic programs at home in the West. It is only "half-way" if you ignore the imperialism and depredations in Africa and Latin America on which it is based. Those who fawn over "Nordic Socialism" while condemning necessary radical, anti-imperialist revolutions in the "Third World" are basically engaging the socialist version of white supremacy.

There seem to be competing definitions and emphasis for the term "democratic socialism," but basically it is a blending. Whether it is as your post phrases it or rather socialists revising capitalism is up for debate or may even be a semantics game.
I think of it as revised capitalism to dampen the power and abuses of private capital through heavier regulation and taxation, which in turn is revenue that can be used to better fund projects and services that benefit the masses. This is hopefully achieved through a democratic process.
Some very significant level of private ownership of means production exists under this. The government "owns" relatively little, but the means of production is taxed and regulated in an attempt to keep it answerable to the needs of the people.

All societies have at least some degree of a blend of private and public ownership of the means of production, except arguably total police states like N. Korea, which has to be propped up economically from without. Even then, you could argue it is a capitalist monopoly of one, which makes categorization rather pointless.

All the rest are some degree of blending, which is one reason why the definitions are not clearly resolved. We have no "pure" ideal examples we can point to as viable economic systems unto themselves.