Include “registration.php” for custom registration form?

If you look at the source of the file you can see that it doesn’t do anything even if you were to include it except add a warning that “This file no longer needs to be included”. So those tutorials are out of date. You don’t need to include it.

The file you actually need to include, if any file at all, depends on what you want to do exactly, and you don’t explain that in the question. A number of registration functions are in wp-login.php which is very inconvenient because you can include that file to get at the functions without also getting a bunch of echoed HTML as well.