One button on admin page works but not two


I am workin on a plugin (for use on my own site). I recently added a button to the admin page that generates some text, and it works fine. This is what I use (pilfered from examples):

if (!current_user_can('manage_options'))  {
  wp_die( __('You do not have sufficient galooph to access this page.')    );
}

if ($_POST('plugin_button') == 'thing' && check_admin_referer('thing_button_clicked')) {
  plugin_thing_button();
}
echo '<form action="options-general.php?page=plugin-list" method="post">';

wp_nonce_field('thing_button_clicked');
echo '<input type="hidden" value="thing" name="plugin_button" />';
submit_button('Generate new thing');

echo '</form>';

This works fine and calls the function as it should.

Now I want a second button to do something completely unrelated.

Here is what I tried, basically copying from above:

if (!current_user_can('manage_options'))  {
  wp_die( __('You do not have sufficient galooph to access this page.')    );
}

if ($_POST('plugin_button') == 'thing' && check_admin_referer('thing_button_clicked')) {
  plugin_thing_button();
}
if ($_POST('plugin_button2') == 'thing2' && check_admin_referer('thing2_button_clicked')) {
  plugin_thing2_button();
}
echo '<form action="options-general.php?page=plugin-list" method="post">';

wp_nonce_field('thing_button_clicked');
echo '<input type="hidden" value="thing" name="plugin_button" />';
submit_button('Generate new thing');

wp_nonce_field('thing2_button_clicked');
echo '<input type="hidden" value="thing2" name="plugin_button2" />';
submit_button('Generate new new thing');

echo '</form>';

Where is my mistake? Thank you in advance!