2 Loops, Only Displaying 1 Loop in Both Loops

You can’t add the pre_get_posts action right before the loop. You HAVE to do it in functions.php. It has to be set before the site actually queries the db to set up the loop which happens before your template file is loaded.

What you are doing is looping through the exact same query twice, because pre_get_posts can’t change a query after happens. In this case, the easiest way to change it is to query_posts to set up a whole new query. Or manually build a query with WP_Query.

The ideal situation might be to use pre_get_posts for one loop and a new query for the other, but a simple solution is something like this:

<div class="section-header clearfix">
    <h2>Movie Trailers</h2>
    <div class="section-filter">
        <ul>
            <li><a class="active" href="http://site.com/videos/">Recent</a></li>
            <li><a href="http://site.com/tag/movie-trailers/">More Movie Trailers</a></li>
        </ul>
    </div>
</div>
<?php 
$movie_trailers = new WP_Query( array(
    'category_name' => 'videos',
    'tag' => 'movie-trailers'
) );
if ( $movie_trailers->have_posts() ) : ?>

<?php while ( $movie_trailers->have_posts() ) : $movie_trailers->the_post(); ?>
    <?php
        get_template_part( 'content-videos', get_post_format() );
    ?>
<?php endwhile; ?>


<?php else : ?>

    <?php get_template_part( 'no-results', 'index' ); ?>

<?php endif; ?>




<div class="section-header latest-gaming-trailers clearfix">
    <h2>Gaming Trailers</h2>
    <div class="section-filter">
        <ul>
            <li><a class="active" href="http://site.com/videos/">Recent</a></li>
            <li><a href="http://site.com/tag/gaming-trailers/">More Gaming Trailers</a></li>
        </ul>
    </div>
</div>
<?php 
$gaming_trailers = new WP_Query( array(
    'category_name' => 'videos',
    'tag' => 'gaming-trailers'
) );
if ( $gaming_trailers->have_posts() ) : ?>
<?php if ( have_posts() ) : ?>

<?php while ( $gaming_trailers->have_posts() ) : $gaming_trailers->the_post(); ?>
    <?php
        get_template_part( 'content-videos', get_post_format() );
    ?>
<?php endwhile; ?>