Prevent reviewer name from converting to "Author" after saving a Word document

Figured it out. Under Prepare → Word Options → Trust Center → Privacy Options in Word 2007 there is a Document-Specific Setting to "Remove personal information from file properties on save"

Unchecking this option fixed the problem. This option is only available for documents that were created in an earlier version of Office and when the option was used in the earlier version to remove personal information. That explains why the issue was occurring with some documents and not others.

In Word 2010, go to File->Options->Trust Center->Trust Center Settings. Then, uncheck "Remove personal information from the file properties on save"

In Word 2013, it falls under File -> Options -> Trust Center -> Trust Center Settings... -> Privacy Options. It is found under the Document-specific settings section.

In Word 2016 for Mac, go to the Review tab on the Ribbon → "Protect Document".  Then uncheck "Remove Personal Information from this file on save".