Yes, this only works if you have Mavericks. It is called Safari Push Notifications because it's a push notification feature provided by Safari, to distinguish it from OS X's more general support for push notifications.

Safari also supports the "Web Notifications" specification[1], which allows a website to post a notification while it is open in Safari. This has been supported for a few releases now and is also available in some other browsers. I believe Gmail can make use of this to call attention to new emails while open in a background tab.

[1]: <https://dvcs.w3.org/hg/notifications/raw-file/tip/Overview.h...