Working from the cable side did the trick for me. Put 3 squirts of 90% alcohol and then worked with a card, pick and spudger on the battery. It took about 5 minutes of gentle prying, but the battery did free itself and no damage was done.
If you are replacing the screen, you will need to transfer the square/triangular board from the old screen to the new screen. This step/detail is missing.
This step does not make sense. In the first picture, he is lifting the motherboard from the base. Then in all subsequent steps is lifting the motherboard from the top. Removing the keyboard/base connector is completely unnecessary unless you have an extremely swollen battery and need the clearance.
Screen also broke on me when doing the repair. Got about halfway down the side of the Surface when the screen cracked in half. I wasn't planning on replacing the screen while replacing the battery and was pretty upset when it cracked. But as someone else said the screen is a "cost" of replacing the battery.