Captions do not display for Images inserted from Media Library

Expected behavior

When you insert Images from the Media Library, and set a caption for them, the caption should show up in the Post when Previewed and Published.

Actual behavior

The captions do not show up in the post when Previewed and Published.

The caption for the Image inserted from the Media Library shows up in the Editor:

But not in the Preview:

Or in the Published Post:

Steps to reproduce the behavior

1. Switch to Media LIbrary and upload a few Images
2. Create a new Post
3. In this post, insert images from the Media Library to the post and add caption. Observe that the caption shows up in the Editor.
4. Hit Preview, then Publish. Notice that the caption does not display in Preview or Published Post.

Tested on iPhone XR / iOS 13.2.3 / WordPress for iOS 13.7 / NOT using block editor

First tested by @melindahelt, and reported by a user in 2370166-zen and 2532609-zen.

Originally reported it on the WordPress iOS GitHub repo - https://github.com/wordpress-mobile/WordPress-iOS/issues/13058 - but was asked to cross-post it here because it's an Aztec Editor issue.

cc @designsimply @yaelirub

Start by reproducing the issue in AztecEditor-iOS with an image from the Media Library, comparing the editor output with Preview and the published post. Trace where the caption is lost between insertion and rendered HTML, then add coverage for the reported workflow; done means captions appear in both Preview and published output.