I have found that the set of books shown by Manage Your Content and Devices as being on each device often disagrees with the actual books present on those devices. Once that happens even syncing does not to fix it. Deregistering a device and registering again appears for fix it for a while.
The Kindle Cloud Reader used to have the ability to download and "pin" books for offline reading. That was recently removed as a feature so that might account for it always showing no downloaded books.
Send to Kindle sends books to a Kindle device or app with the option to have them saved to your account for future downloading. Those saved "personal documents" appear in Manage Your Content and Devices under "Docs". Kindle Cloud Read (along with with Kindle for Mac/PC) will not show personal documents.
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