The bookmarks I want to save, all organized just the way I want them, are on my iPad. I want my MacBook Air to have this same set of bookmarks.
So you need to do it kind of backwards. Never done it that way. I'm assuming that you have iCloud Safari syncing switched OFF in your MacBook and also OFF in your iPad. Here's what I'd do…
• Delete all bookmarks from Safari on your MacBook Air.
• Switch on iCloud Safari on your MacBook Air only.
• If bookmarks reappear on your MacBook Air (from iCloud), delete them.
• Continue until you can leave it overnight, switch iCloud Safari OFF and then back ON until no bookmarks reappear on your MacBook Air. The idea is that both iCloud and MacBook are empty, and synced to each other.
• Take a deep breath and pray.
• Switch on iCloud Safari syncing on your iPad.
Ideally, the bookmarks from your iPad will propagate into the empty iCloud, and from there to your empty MacBook Air, and all live happily ever after.
• Continue to pray for a day or two that your iPad and/or Mac does not get inundated with bookmarks that were hiding in a dark corner of iCloud.
I back up my MacBook using Time Machine and a WD Passport external drive. I'm thinking of turning off Time Machine and clearing it and also wiping the Passport.
That will not help with this issue. You should only do this if you're trying to resolve some other issue.
I came to the conclusion that it might be coming from my backups.
No, I disagree. I've never seen TIme Machine restore stuff automatically. If you had iCloud Safari switched on during this time, it was probably the work of iCloud.
I do not have the software you recommend.
The non-Apple software is not necessary. It is only handy for safe-keeping bookmarks while you're cleaning things out on your Mac. In your case, since the bookmarks you want to keep safe are on your iPad, it would not help you at all.
Couldn't find apps to organize bookmarks for iOS 7.
That's because there are none. Apple does not allow any non-Apple app to access with Safari bookmarks on iOS.
I just don't understand how iCloud knows which bookmarks I want to keep. I feel like most people are wanting to keep the bookmarks on their Mac. I want to keep the bookmarks on my iPad. How does iCloud choose?
It doesn't. It merges bookmarks from all devices together and keeps everything.from everywhere. When in doubt about anything, it keeps that too.