Some of you need to shut the hell up with "Democrats according to European standards would be center-right". We know but this isn't Europe.
Also, the Democratic Party has a wide range of views. The Democratic Party has a bigger ideological tent than the Republicans by far. Finally, the Democratic Party shifted to the right due to the rightward shift of the country (due to racist and "racially anxious" White voters after the Civil Rights Movement).
Only one Democrat was elected from 1968 until 1992. Ronald Reagan's presidency demonized government-funded intivaites and programs as "socialism/communism" and portrayed the Democrats as being "weak on crime" (you guys already know what that means).
In 1992, the Democrats had lost 3 straight presidential elections in a row (and all by decent margins too). Democratic voters nominated a young and charismatic governor who was more to the right than previous Democratic nominees.....and it worked. Billy the Comeback Kid won....and then he was re-elected. A Democratic president hadn't been re-elected since FDR in 1944.
The Democratic establishment was going to adopt "Third Way" as a result to trying to keep up with the Republican Party. The Democratic establishment should abandon "Third Way" politics for good in this day and age though.
Exactly, all I've been trying to say

I will say that the tension we are seeing in the party today is a direct result of that rightward shift. Which is why for me, labels like "center left" have little meaning without the context you accurately stated. And long-term this tension has to be resolved one way or another. Either that rightward drift stops, or voters disillusioned with having to choose between a ultra right or "center left" (actually center right) platform will sit out more and more.