This requires delicate handling of type definitions which collide, because
we use TypeScript-provided lib.d.ts for --target=es5 and lib.es6.d.ts for
--target=es6.
We need to include our polyfill typings only in the --target=es5 case,
and the usages have to be consistent with lib.es6.d.ts.
Also starting with this change we now typecheck additional modules,
so this fixes a bunch of wrong typings which were never checked before.
Fixes#3178
