OK-- sounds like choice #2. Is your drive Solid State? SSDs are way faster and more satisfying. Some mechanical drives seem very slow. I copied files to a 10GB spinning drive last night for archival storage, and it took forever! This would be awful for a Photos Library.
Two odd things about running a Library off an external drive: (1) if you unplug the drive, it could cause a major re-sync that takes a while to stabilize. If need to travel or something, I'll just turn off the WiFi, unplug the drive, then plug the drive back in and finally turn on the wifi again. I haven't actually seen any problems. (2) location maps seem to sometimes be confused-- the location is recorded properly, but the map may not work on recent pictures. I hope this will be fixed.
You created a new Library on the external drive, is that right? If so then double click on it to open it-- you should see an empty Library. Then do Settings>General to see this:

Double check the path to be sure you are working with the right Library. In this picture "Use as your System Library" is grayed out, because it already is the system library. Only the System Library can connect to iCloud. So if your "Use as" isn't grayed out, then click on it to set it as the System Library.
Be sure that Importing has "Copy Items" checked.
Go to Settings>iCloud and connect this new Library to iCloud.

Set "Download Originals," since you have the storage space available on your drive.
Then wait. Wait some more, and eventually this new Library will fill with full sized images from iCloud.
Make sure this Library has all your pictures, then you can trash the old Library.
Is this doesn't work for you, let us know where it goes wrong and what you see…