Clarification of Definition of iMessage

Indeed. End-to-end encryption protects your iMessage and FaceTime conversations across all your devices. With watchOS, iOS, and iPadOS, your messages are encrypted on your device so they can’t be accessed without your passcode. iMessage and FaceTime are designed so that there’s no way for Apple to read your messages when they’re in transit between devices. You can choose to automatically delete your messages from your device after 30 days or a year or keep them on your device indefinitely. If you want to learn more about the privacy of iMessages, open Settings, select Messages, and scroll all the way down and click on “About Messages for Business & Privacy”. They also happen automatically as long as it’s turned on. You can also view Apple’s privacy practices at Privacy - Apple

They are "encrypted end-to-end" and nobody but the sender and receiver can read them, not even Apple.

Yes, it happens automatically between Apple devices unless you shut it off in your settings (Settings, >Messages, the toggle to off)