GH-97950: Use new-style index directive ('builtin') (GH-104164)
* Uncomment builtin removal in pairindextypes
* Use new-style index directive ('builtin') - C API
* Use new-style index directive ('builtin') - Extending
* Use new-style index directive ('builtin') - Library
* Use new-style index directive ('builtin') - Reference
* Use new-style index directive ('builtin') - Tutorial
(cherry picked from commit f5088006ca)
Co-authored-by: Adam Turner <9087854+AA-Turner@users.noreply.github.com>