You're not wrong but that's the German definition.

On the Wikipedia page of "Three Arrows" in the Section "Use outside Germany" at the end of paragraph 2 it says:

"After World War II, the Three Arrows became the official party logo of the Social Democratic Party of Austria (SPÖ) in 1945. The symbol had been modified to include a circle, and the symbolism changed to represent the unity of industrial workers, farm workers and intellectuals. The Three Arrows remained a prominent symbol of the Social Democratic Party of Austria until the 1950s. According to the modern SPÖ, the Three Arrows represent opposition against fascism, capitalism and clericalism."

As to why they changed the definition my guess is the Austrian Social Democrats didn't want to oppose communism because the first republic was essentially a two (or at best times three) party system similar to the US before it drifted into Austro-Fascism and a civil war (foreshadowing for the US) and ultimately got annexed. And until today the SPÖ is more left than the SPD - or should I say their left wing is stronger.