set widget name text

I’m writing a multiple-instance widget using the widget_many template provided in widgets.php. All is well, except I can’t seem to mimic the behavior of the Text widget (for example) that displays the $title variable as part of the widget name on the admin page.

If I set it just before wp_register_sidebar_widget like so:

if (!(empty($title))) $name = __('My Widget :' . $title);

…it works, but then the first instance of My Widget: title also become the name for the generic draggable “model” widget on the left side of the admin screen.

I’m following the code to text-widget pretty much exactly, but can’t find where it changes the $name of each widget instance!