List of authors posts minus very latest

There’s no need for multiple queries – posts are, by default, ordered by most recent. Just apply a classname for the first post on the first page:

<article <?php post_class( $wp_query->current_post === 0 && ! is_paged() ? 'my-awesome-first-post' : '' ) ?> 

The first post of the first page will have the class my-awesome-first-post.

Update: Based on your updated question’s code:

<?php while ( have_posts() ) : the_post() ?>

    <div <?php post_class( $wp_query->current_post === 0 && ! is_paged() ? 'first-post' : '' ) ?>>
        <h2><a href="https://wordpress.stackexchange.com/questions/214644/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h2>
        <?php the_post_thumbnail();?>
        <small><?php the_time('F jS, Y') ?> by <?php the_author_posts_link() ?></small>

        <?php the_excerpt() ?>
    </div>

<?php endwhile ?>