I figured out why the simple captcha is not working.
wp-login.php calls:
wp-load.php calls:
wp-config.php calls:
wp-settings.php which invokes:
wp_unregister_globals() (around line 46) which unsets $_SESSION variables if register_globals is ON.
If register_globals is turned OFF on your PHP server, then simple captcha will work. This explains why it works for some and not others. BTW register_globals is OFF permanently in PHP 6.0.