Thank you very much. I had some trouble with the home button (not reassembled properly), but on the second trial, it worked as described.
Not related to the tutorial, but to the phone itself: I did have problems with the Pentalobe screws at the bottom of the phone (step 1) - I was not able to unscrew one of them, I had to “force” the screen out of the case with one of the screw still in place. It worked.
It worked smoothly! Less than 15 minutes. I used a OEM-battery, instead of an original one.
Took me less than 10 minutes to complete. Computer rebooted as soon as put the MagSafe connector. I had to force a synchronization of the date/hour in Preferences to get it up-to-date (I guess the system would have done automatically a little bit later).
The battery came from iFixit. Battery stats, as shown by Battery health https://fiplab.com/apps/battery-health-f... show now:
Original max: 4680 mAh, Current Max: 4758mAh; Health: 102%, OS X Health Status: Good
The battery came is reported to be 3 years old(!) and came at about 80% charge
Instructions are very clear!
I think I found the way to have the new battery to charge!!!!!
Essentially, you have to follow a very precise order: 1/shut down the computer, 2/take off the magsafe connector, 3/remove the old battery, 4/put back the magsafe connector, 5/put the new battery, 6/power on, each time leaving a few seconds in between.
I just replaced the battery as indicated (with a iFixit part), but the computer says there is no battery connected, even after letting the power plug in for more than one hour.