“AVOID using library private types in public APIs” – lint warning, even in in cookbook examples?

From the latest docs:

Subclasses should override this method to return a newly created
instance of their associated [State] subclass:

@override
State<MyWidget> createState() => _MyWidgetState();

So you should replace

@override
_AnimatedContainerAppState createState() => _AnimatedContainerAppState();

with

@override
State<AnimatedContainerApp> createState() => _AnimatedContainerAppState();