How to handle “the_terms” inside loop

To answer your first question

What is the difference between those functions

• get_terms() returns an array of terms objects that belongs to a specific taxonomy

• get_the_terms() returns an array of terms belonging to a post

• the_terms() displays an HTML formatting string of term names belonging to a post

Because you need your terms not hyperlinked and ordered according to parent, I believe wp_get_object_terms() will be a better option here. wp_get_object_terms() also returns an array of terms belonging to a post, but is more flexible. You do pay for this flexibility though as you make an extra db call per post.

With this all in mind, you can try the following: (All code is untested)

$args = [
    'orderby' => 'parent', 
    'order'   => 'DESC' 
];
$terms  = wp_get_object_terms( $post->ID, 'locations', $args );
$names  = wp_list_pluck( $terms, 'name' );
$output = implode( ', ', $names );
echo $output;