Right, this is really working on my nerves, but Instagram has to do something about their bloody documentation.
I am already trying for a week to live update my website
The Instagram documentation is accurate and actually well written.
The realtime API is working correctly. As stated in the documentation:
The changed data is not included in the payload, so it is up to you how you'd like to fetch the new data. For example, you may decide only to fetch new data for specific users, or after a certain number of photos have been posted.
http://instagram.com/developer/realtime/
You only receive a notification that an update has happened to your subscribed object. It is up to you to call the API to find out what that data is.
You can call the /tags/[tag-name]/media/recent
with an access token that you have previously stored on your own server or DB. Then, you should be able to compare the data returned from that endpoint with any data you have retrieved prior, and just pull the objects that you do not yet have.