The guide says: "Grab the clear plastic pull tab attached to the battery connector and pull it toward the front edge of the Air to disconnect the battery from the logic board", but the battery model that I was replacing did not have a clear plastic pull tab at all...
The battery model shown in this guide is: A1405
I was installing battery model: A1406
A1406 has the following connector:
Top:
http://www.directupload.net/file/d/4122/...
Bottom:
http://www.directupload.net/file/d/4122/...
By looking at the bottom of the new battery I saw how it could be removed. Simply by a gentle upward movement. I very, very carefully nudged it a bit upwards using a screwdriver on the left and right side, as shown here:
http://www.directupload.net/file/d/4122/...
It came out fairly easy, luckily!
(And yes, I also cleaned up this MacBook before closing it again. It was a bit hairy in a literal way. Electronics and pets is not the best combination. ;-) )
This was a great guide!
