Two things ...
1) The quote from Zend says DON'T do "Something <Something>", because it might be parsed incorrectly by the server.
2) I can't see anywhere on the manual reference where it says to include the "To: " bit at the front. I think you're getting confused with the format of the 'EXTRA headers' parameter.
Standard usage:
$to = "fred@bloggs.com"
$subject = "Subject";
$message = "Blah, Blah";
$extra = "From: webmaster@".$_SERVER['SERVER_NAME'],
mail($to, $subject, $message, $extra);