[RN] Move definition of public instances to ReactNativePrivateInterface (#32446)

## Summary

> [!NOTE]
> This only modifies types, so shouldn't have an impact at runtime.

Some time ago we moved some type definitions from React to React Native
in #26437.

This continues making progress on that so values that are created by
React Native and passed to the React renderer (in this case public
instances) are actually defined in React Native and not in React.

This will allow us to modify the definition of some of these types
without having to make changes in the React repository (in the short
term, we want to refactor PublicInstance from an object to an interface,
and then modify that interface to add all the new DOM methods).

## How did you test this change?

Manually synced `ReactNativeTypes` on top of
https://github.com/facebook/react-native/pull/49602 and verified Flow
passes.