What is the best way to load the dependencies for the
WP_Filesystem? The first or the second code part?
The second one is good and I would use that as well.
And WP_Filesystem() must be called so that the global $wp_filesystem variable is properly initialized/setup, i.e. assigned with the proper class or file system method (see get_filesystem_method(), which defaults to WP_Filesystem_Direct), and therefore allowing you to use the WordPress API for file/directory operations such as checking if a file exists using $wp_filesystem->exists() or getting the file content using $wp_filesystem->get_contents().
Example:
global $wp_filesystem;
// Make sure that the above variable is properly setup.
require_once ABSPATH . 'wp-admin/includes/file.php';
WP_Filesystem();
// Check whether a file/directory exists.
$exists = $wp_filesystem->exists( '/some/path/here' );
var_dump( $exists );
// Get file content.
$data = $wp_filesystem->get_contents( 'path/to/file' );
var_dump( $data );