How to conditionally include a custom field on category editor screen not category “add” screen

You can target edit-tags.php page and additionally check for that edit action.

add_action( 'admin_head-edit-tags.php', 'my_category_edit' );

function my_category_edit() {

    global $action, $taxonomy;

    if( 'edit' != $action || 'category' != $taxonomy )
        return;

    // code goes here
}