The purpose of $page in node.tpl.php and when does it become FALSE?
That variable becomes false in:

template_preprocess_node

with

$variables['page'] = $variables['view_mode'] == 'full' &amp;&amp; node_is_page($node);


See also:

node_view

For the view_mode parameter used there.

The output of the node $title is controlled by page.tpl.php for the full page node view.