How to change the product short description for each product with PHP?

What you are looking for is the excerpt (also called short description) and it’s not stored as a meta value. It’s stored with the post in the post_excerpt column.

So you can try something like this :

// Update product with id = 44
  $my_post = array(
      'ID'           => 44,
      'post_excerpt'   => 'New text',
  );

// Update the post into the database
  wp_update_post( $my_post );