Is the $_SESSION['name'] variable an array? If you want to delete a specific key from within an array, you have to refer to that exact key in the unset() call, otherwise you delete the entire array, e.g.
$name = array(0 => 'a', 1 => 'b', 2 => 'c');
unset($name); // deletes the entire array
unset($name[1]); // deletes only the 'b' entry
Another minor problem with your snippet: You're mixing GET query parameters in with a POST form. Is there any reason why you can't do the forms with 'name' being passed in a hidden field? It's best to not mix get and post variables, especially if you use $_REQUEST elsewhere. You can run into all kinds of fun trying to figure out why $_GET['name'] isn't showing up the same as $_POST['name'], because the server's got a differnt EGPCS order set in the 'variables_order' .ini setting.
<form blah blah blah method="post">
<input type="hidden" name="name" value="<?= htmlspecialchars($list1) ?>" />
<input type="submit" name="add" value="Add />
</form>
And note the htmlspecialchars() call. If either $list1 or $list2 contain a double quote ("), it'll break your HTML