A document will go to Under Signing by Author status if a reviewer or reviewers have made changes to a document and when all reviewers have signed-off on the same version.
For a Document to go to Under Signing by Author status it means there has been an update by one or all Reviewers. In other words the last Reviewer(s) have signed off on a different version than the Author signed off on while the Document was in Draft status.
When any Reviewer signs off on a Document, the text will always read 'Set to Under Signing by Approver'.
In addition, if there has been any version increase during "Under Review" the Reviewers who signed off before the version update will need to sign off again as will the Document Author or Authors.
This is to ensure that the Document will only go to the Approver if no changes have been made and only once the last Reviewer signs off, (unless you have configured that all Reviewers need to sign off).
The requirement in any document approval workflow is that everyone signs off on the same version of the Document. Scilife cannot control the changes made to the Document while editing in Word, what it can control is the version number and ensure that everyone signs off on the same version.