I would like to make a minor change to the xoopsgallery_index.html file located in modules/xoopsgallery/templates directory. Modifying the file directly does not work - smarty (re)generates the old file (as inspected in the templates_c directory). Under Administration/System Admin/Preferences/General Settings, I checked "Yes" for the "Update module template ..." setting and placed the modified file under themes/mytheme/templates as instructed, but I still cannot get the changes recognized. What am I doing wrong?

I think you have to modify the template directly from the admin panel. Go to the template management section on admin panel and make a copy of the default template set. Edit the file you want to change (on your copy template set). At the main preferences setup, change the site template set to the one you have just created. It should work.

The setting "update module templat.." will update THEME files automatically if changed.
To have the templates in e.g. modules updated after a modification, you should update the module in system admin -> modules admin

yes... and if you don't want lose your modif, you have to change the file in the /module/yourmodule/template else you lost your change when updating the module via the admin panel.