disable password protected page for logged users

Disable the post password protection for (some) users

You can try the post_password_required filter (4.7+) to override it for logged in users:

add_filter( 'post_password_required', function( $returned, $post )
{
    // Override it for logged in users:
    if( $returned && is_user_logged_in() )
        $returned = false;

    return $returned;
}, 10, 2 );

or disable it for users with a given capability ( e.g. manage_options) :

add_filter( 'post_password_required', function( $returned, $post )
{
    // Override it for users with the 'manage_options' capability
    if( $returned && current_user_can( 'manage_options' ) )
        $returned = false;

    return $returned;
}, 10, 2 );

We could target a given post type with:

add_filter( 'post_password_required', function( $returned, $post )
{
    // Target protected posts only
    if( ! $returned )
        return $returned;

    // Target logged in users only
    if( ! is_user_logged_in() )
        return $returned;

    // Target 'page' post type only
   if( 'page' !== get_post_type( $post ) )
        return $returned;

   // Override 
   return false;

}, 10, 2 );

Remove ‘Protected’ text from protected post titles

To remove the prepended Protected from the title, we can adjust the protected title format, for logged in users, with the protected_title_format filter:

add_filter( 'protected_title_format', function( $format, $post )
{
    if( is_user_logged_in() )
        $format="%s";

    return $format;
}, 10, 2 );