Are you using "Optimize Storage" on Mac #1? If you're not using Optimize, then you can copy the Photos Library directly to an external drive and then to Mac #2. Alternatively, you can use Target mode to connect the two Macs directly with a cable, and then you don't need the intermediary drive.
Transfer files between two Mac computers using target disk mode - Apple Support
Again, it's the Photos Library, in the Pictures folder, that you copy, not "download" pictures.
If you are using "Optimize Storage" on Mac #1, then all your picture originals are not on the Mac, and copying the Library would not be complete. In that case, your best bet would be to make a new user on Mac #2, say User#2, sign in to Apple ID #1, and connect Photos there to iCloud. Wait a week or more for all the pictures to download, and then transfer the Photos Library from user#2 Pictures folder to user#2's Public folder. Then, back to the original user on Mac #2, transfer the Photos Library from user#2's Public folder to the original user's Pictures folder.
Let us know what you're doing and if you have more questions. (My #1 and #2 things might be a little confusing…)
