Restore photos library from external hard drive to new Mac

Hi,

I received instructions on how to restore a library from Time Machine (specifically my Photos library) from an external hard drive to my new Mac. I don't want to do a complete transfer of all data - just specifically the photos library.

Anyway, I have connected my external hard drive and followed these instructions:

Restore a library from Time Machine in Photos on Mac - Apple Support

However, there is NO folder called photo library or photos in my time machine backup, if I enter through my external hard drive.

If I enter time machine through my new Mac, it just says iCloud is full and there is nothing to select to restore.

What am I doing wrong?