What's the correct way to back up all Photos from the Photos app? I downloaded the macOS Sierra (GM) and installed it on my MacBook. Before I installed it a backed up all my data that isn't stored in the cloud or somewhere else.

I'm using iCloud Photos to sync all my photos and videos with iCloud. Every photo and video is already in iCloud so I may don't need to back up my Photos library, but I did - just in case something goes wrong.

So I wiped off all the data from my MacBook to do a complete and clean installation of Sierra. Finally everything works fine and I'm going to get all my photos and videos back to the Photos app.

My backup of my photos and videos has ~13GB and I really have a slow internet connection, so it would be nice if I could use the backed up library instead of downloading everything from iCloud.

As soon as I open my library with Photos it starts importing and optimizing etc. and all photos and videos are there. But it starts uploading every single photo and video to iCloud? How can I tell Photos that my photos and videos are already in iCloud and it should stop uploading?
The Photos app in macOS Sierra is able to do a comparison between what is in the cloud, and what is stored in the local library on your Mac. (Likely through some means of signature/hash function).

To provide some insight with actual numbers - My Photos library is approximately 53GB, and only required about ~1.5GB worth of bandwidth to validate/sync the library after migrating the library to a different Mac with a fresh install of macOS Sierra (1.5GB includes any photos that it also needed to sync since the last my library was active). 

With that being said, you can be confident that your bandwidth limitation, or capacity, should not have to handle the full file size. However, be mindful that if there are other cloud services (iCloud Drive) they too can utilize bandwidth.