Configuring e-signature display

Hello,

I have Adobe Acrobat Pro with an Enterprise license.  When I send documents for e-signatures using the Request E-Signature feature, I would like to be able to configure how the signatures are recorded.  Specifically, when someone signs with the text option, it prints their name as well as a timestamp with the date and time.  If they sign with an image or drawing the signature, it gives only the date (and sometimes not even that, if they have saved the signature).

Is there a way to force the e-signature system to print the name and timestamp with every signature?  We have digital signatures (and it would be great to be able to e-signature with them!) but we need to do this with e-signatures so we can keep the audit logs.

(Initially I posted this in the Acrobat forum: https://community.adobe.com/t5/acrobat-discussions/configuring-e-signature-display/m-p/13435991#M391747)