liberals often use the nebulous, ill-defined term of “Democratic Socialism” as an AES cudgel.

I see liberals try to equate any kind of public sector combined with a national election system as Democratic Socialism. Which gets you the Nordic Model - a collection of petrostates with an egalitarian veneer and a white supremacist underbelly - labeled "Democratic Socialism" on paper.

Meanwhile, actual social democracies in Latin America, Africa, and East Asia are denounced as authoritarian every time the Neoliberal (or outright reactionary) local politician loses an election.

I am stating that AES is democratic as is Marxism in general

Marxism is Democratic in theory. Leninism is more popular than democratic, as Leninists aren't wedded to electoralism like their liberal peers.

But the critique I see most often among liberals is that markets are democratic. And therefore every AES state that fails to sufficiently privatize the economy is definitely facto authoritarian.

That's the real definitional divide between Marxists and Liberal Democrats.