Difference between impurity and misclassificaton
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.