Change Login or Logout text based on status

You have the PHP IF statement needed, is_user_logged_in().

Just add your button to that and you should be good. Something this…

<?php if (is_user_logged_in()) {
    echo '<button type="button" class="show-btn">LOG IN</button>';
} else {
    echo '<button type="button" class="show-btn">REGISTER</button>';
}; ?>

Note: You have an extra </div> in your code above.