Replace SSD without unplugging laptop battery

Solution 1:

Generally swapping a SATA-based drive with power on will not damage the hardware, as SATA drives are inherently hot-swappable (the controller may not be though, depending on age and settings). Even when hot swapping is fully supported, it still may cause data loss or corruption, depending on what the drive is doing when you pull it. But as long as you have a backup, you shouldn't lose anything. ;)

"The heads (Philips) got a bit damaged by my attempt"

Perhaps get a set of stripped screw extractors and remove the screws properly (and carefully).

Otherwise, use the provided emergency power cut-off. Changing the SSD is not "normal conditions", and they are referring to hitting it while the computer is on and functioning normally -- much in the same way you wouldn't pull the power cord to a desktop PC from the wall under "normal conditions", as it may cause corruption and data-loss.

"I'm not to keen on testing if it works..."

If you are afraid to try things because it might wreck something, then you should probably pack it up and take it to a qualified service depot so they can bear that responsibility.