I understand that killing power to a running pi without first shutting down can damage the SD card.
Can Killing power at the GDM3 login screen also damage my SD card or do I need to run the GDM3 shutdown function before killing power.
Short Answer: Just as likely as pulling the plug on an idle system at any other time. This risk is lower than when the device is actively using the SD card. Whether that risk is acceptable is up to you.
The likelihood of physically damaging an SD card from hard power off is rather low and usually requires additional extenuating circumstances.
What tends to happen is file system corruption if the power is pulled during an interaction with the card.
Most file system operations are buffered by the kernel. The kernel schedules synchronization at convenient time, so there is a small chance of an ongoing file system operation at any time.
When you are at the
GDM boot screen, the system is completely booted, any services you are using are already running, there is no logged in user, but most of the system functions are operating the same way they are at any other time.