Migrating outdated Photos library from iCloud to new Mac Mini

Pictures synced to iCloud Photos retain keywords, even though they are not visible there. So there's that!

You can have more than one Library on a Mac, so if you connect an empty Library to iCloud, you can still do what you want with the old Library. I'd turn off iCloud on the old MacBook; then I would connect the MacMini to iCloud and see what you get.

Here's a crucial question: And how is your MacBook's external drive formatted? Select the drive and do ⌘-i to open the info window:

To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:

Move your Photos library to save space on your Mac - Apple Support

If the drive is not properly formatted, then you shouldn't run the Library from your Mini.

After making sure that the external drive is properly formatted, I'd make another copy of the Library, putting in a different folder. You didn't say, but I'm guessing that the ten year old MacBook is running an older operating system. If that's so, then when you connect your Mac Mini to the old Library, it will want to update the Library to the new OS, and without that, it can't work on the Mini. And once it's updated, it will no longer work on an older OS. So you should keep a copy of the old Library that is unchanged. Backups are always good to have. You might name it something like "Old Library.photoslibrary" so you can tell it apart from the new one.

If you connect this (properly formatted) drive to the MacMini, then you can double click on the old Library and it will open in Photos after an update. You can compare the pictures.

If you want to copy pictures from the old Library to the new one, you can use the trusted 3rd party app PowerPhotos ($30). This makes working with multiple Libraries much easier.

Let us know any questions you have, and how it's going…

Which system version is running on your old 2014 MacBook Air?

My oldest Mac with iCloud Photos is running macOS 10.14 Mojave. I had not turned it on for a long time, and when I turned it on again to access videos with an older codec, I thought that iCloud syncing did no longer work, after I upgraded my newer Mac to Sequoia. But I had just been not patient enough. After keeping the older Mac up and running for a very long time, I could see, that all my photos and videos were still there, and have been downloaded from iCloud Photos.

Only the photos I have moved to the Shared iCloud Photos Library cannot be accessed on the older system versions. Are you using the Shared iCloud Photos Library on your new Mac?

I would not wipe your old external drive with the old iCloud Photos Library. I would keep it as an additional backup. You may need a version of your Photos Library in a format that the old Mac con open. I found the old versions of my Photos Libraries very useful as a backup, when I am coming across legacy videos that will no longer play or corrupted JPEG files, where a transmission error has flipped a bit and the JPEG needs to be restored. Older versions of the library on an older Mac are very useful, when we need to convert old videos to a more compatible format or when we need to recover the original version of an image as imported. If a corrupted image is undiscovered for a long time, the only version on Time Machine may also already have become corrupted. Corrupted Originals, the Insidious Danger - How to Protect our Photo Libraries - Apple Community