Will my organization system (folders, albums) remain intact if I move my photos library to an external HD?

1. Make sure the external is formatted Mac OS X Extended, Journaled or apfs.
2. Drag the Photos Library from the Pictures Folder to the external. Leave it time to copy - and it is a copy, your library on the internal is untouched in this process.
3. After the copy is finished - and this can take some time depending on the size of your library - hold down the option key and launch Photos. From the resulting dialogue choose the library on the external. Is everything as you expect it to be? If so, then job done.
4. If this is a move - that is, you're now going to delete the library on the internal - remember you need to back up this library on the external, probably to another external.

If you move the library as a unit to a compatible external drive, the folder/album organization should be transferred intact.

Just to clarify: you say in your original post that you want move the library. Now you mention a "back up to the back up", so I'm not clear exactly what you're doing. I assumed from your post that you wanted to move the library, but if you're backing it up I would suggest alternative ways to work. Can you clarify?

I believe you were suggesting that after I move the photo library, it would be smart to back that new library up, somewhere.

Yes, exactly.

I think the best and safest is another disk. iCloud is not a back up, it's a sharing service. That means there is an outside risk of deleting material inadvertently. I don't recommend Carbonite (or similar services) for two reasons. First, the Photos library must sit on a disk formatted Mac OS X Extended, Journaled or apfs, otherwise it is likely to suffer silent ongoing corruption, risking dataloss. I don't know of any online back up service that offers either of those formats. Secondly there is also the issue of data transfers. Your Photos library can grow to be a very large installation - often hundreds of gigabytes. In the event of a catastrophe, how long would it take to download an item of that size?

That said, uploading your images to a service like Flickr, while not being a back up of your Photos Library, does give you an ultimate failsafe of a back up for your images, only.

So therefore I think that the best back up is a second HD. And for convenience I suggest using incremental back ups. There are apps that do this. The first time you run the back up the app will make a complete copy of the Library. Thereafter it will update the back up with the changes you have made. That makes subsequent back ups much faster. Many of these apps also have scheduling capabilities: So set it up and it will do the back up automatically.

Example of such apps: Chronosync - but there are many others. Search on MacUpdate or the App Store