I sourced all the parts from ifixit, plus a magnetic project mat which I found to be very useful for organising the teardown and reassembly.
In my case the screw points connecting the old battery the case had fractured or broken. The broken bits needed to be cleaned out of the case before installing the replacement battery. Didn't want any loose fragments rattling about inside.
I performed this on my MacBook Air (11-inch, Mid 2011), model A1370. Rather than using the spudger to un-attach the batter connector, I gripped either side with my fingers as the second photographs indicated, and gently pry/wiggled it up and out by initially pinching into the side gap with my finger nails. The reason I did it this way is that the spudger works by leverage and I was afraid to rest it against the components surrounding the connector. The photos and instructions are excellent and illustrate the maneuver perfectly.