Difference between impurity and misclassificaton I am reading the gini index definition for decision tree:  

Gini impurity is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it was randomly labeled according to the distribution of labels in the subset. 


This seems to be the same as misclassification. Is Gini index just a fancy name for misclassification? Or is there really some subtle difference? Thanks!
Is Gini index just a fancy name for misclassification?


No.

Note that Gini index definition doesn't involve predicted values, and also it involves some probabilities, which are not dependent on classifier.

Also in context of decision trees, Gini impurity corresponds to each region, and is not a single value, such as missclassification rate (technically you could also count missclassification rate per region, but then you'd also ).

See this notebook for a concrete example.