switch statement for taxonomy content

Your code is all upside down and scrambled. What you would want to do is to check for a specific term and then feed that into get_the_term_list(), not the other way around. Also, evaluate your switch to true.

I would probably first get an array of terms attached to the post and then check if my desired term is in that array to make my statement a bit more efficient. For this, I will use wp_get_post_terms() and only get the term names as it seems if this is what you are testing against

With this being said, you can try something like this; (CAVEAT: Untested)

$terms = wp_get_post_terms( $post->ID, 'TAXONOMY NAME', array( 'fields' => 'names' ) );
switch (true) {
    case in_array( 'John Doe', $terms ):
        $name="john";
        break;
    case in_array( 'Mark Webber', $terms ):
        $name="mark";
        break;
    case in_array( 'Jane Smith', $terms ):
        $name="jane";
        break;
    default:
        $name="author";
}

echo get_the_term_list( $post->ID, 'TAXONOMY NAME', $name, ', ' );