You mentioned - Messages on iPhone and iPad are showing in iCloud but not those on my MacBook.
messages are synced only with iCloud , they are saved in devices Mac / iPhone , you will not see a message icon when Apple ID is signed in at www.icloud.com
The syncing happens when messages box is checked , the messages are only synced with iCloud , they are not literally seen at iCloud.com
When you turn back messages in Mac , messages will sync only when message box is checked in iPhone , yes those messages will again restored back essentially from iPhone and not from iCloud .
See the articles If you can't send or receive messages on your iPhone or iPad - Apple Support
Forward SMS/MMS text messages from your iPhone to other devices - Apple Support
Change iMessage settings in Messages on Mac - Apple Support
Sync Now
Update your Mac so that all your texts from your other Apple devices appear.
When Enable Messages in iCloud is turned on, your Mac will automatically be updated. However, if you want to make sure you’re seeing the most up-to-date messages on your Mac — for example, if you just switched to a new Mac — you can click Sync Now.
Note: You can also choose this option in iCloud settings. See Change iCloud settings.