Just an update. The module can be checked in, published or unpublished from the summary page of the Joomla module manager. So the problem is definitely related to the module edit page itself.
It doesn't matter whether changes have been made or not. When the Save, Save & Close, Save & New or Close buttons are pressed, the connection is reset every time.
I also asked my hosting service to look into it and they did, but could not find any event in the logs to explain why the connection resets when one exits from the module edit page.
As I said previously, there is nothing keeping me from updating the module fields directly in its MySQL record, just not such a fun process to update all the slide details crammed into the parameters field that way.
Since I'm auditioning free Joomla resources for an article I'm writing, this is a deal killer on mentioning your templates. Sorry
But I hope my careful observations have been helpful.
- Jeff