> These are actually considered equivalent in modern political discourse
Not true. Especially not in Europe.
> hit that Merriam-Webster link in the parent
Unfortunately, Merriam-Webster has it wrong in this case. Social democracy is not a synonym for democratic socialism. The two are very different ideologies with very different goals. As stated in the Wikipedia article for social democracy (with accompanying references):
"Social democracy is a political, social and economic ideology that supports economic and social interventions to promote social justice within the framework of a liberal democratic polity and capitalist economy... Social democracy became associated with Keynesian economics, state interventionism and the welfare state, while abandoning the prior goal of replacing the capitalist system (factor markets, private property and wage labor) with a qualitatively different socialist economic system... By the post-World War II period, most social democrats in Europe had abandoned their ideological connection to Marxism and shifted their emphasis toward social policy reform in place of transition from capitalism to socialism." [emphasis added]
Contrast this with the article for democratic socialism:
"Democratic socialism is further distinguished from social democracy on the basis that democratic socialists are committed to systemic transformation of the economy from capitalism to socialism, whereas social democracy is supportive of reforms to capitalism... As socialists, democratic socialists believe that the systemic issues of capitalism can only be solved by replacing the capitalist system with a socialist system—i.e. by replacing private ownership with social ownership of the means of production."
In summary, social democracy grew out of a movement that advocated transitioning from capitalism to socialism, but ultimately rejected such a transition in favor of working within a capitalist economy. Democratic socialism, on the other hand, advocates for the abolition of capitalism and private property. It does not reflect the present policies of Denmark and Germany, given that neither intends to abolish private property and transition to a socialist economy.