At this point I have no explanation for what you are seeing. I followed the exact same set of steps on my server and the resulting Photos Library is an exact duplicate of the Library on my Desktop — the only connection between the two being their mutual connection to the iCloud Photos Library. What you see on your Laptop and Desktop can be identical but will never be a 100% match to iCloud or iOS simply because MacOS supports features that are not available on the other platforms. But those differences are minor and the images and albums should be the same.
Once again I urge you to CAREFULLY check your Photos and System Preferences on both your laptop and desktop to verify they match my recommended settings as well as each other then for the time being leave both machines plugged in and logged onto the network for a day or two with Photos open on both. After that If the Desktop and Laptop do not reach consensus on the Library organization and content, manually replacing the Photos Library on the Desktop with a copy of the Photos Library on the Laptop. But my recent test proves that should NOT be necessary.
