How to specify if the module’s output is suppressed or not?
Maybe that's the wrong question to ask. If you think about what the user is really attempting to do and then reword it to ask "How do I output the module data to a variable?", then the answer becomes more clear. Just use output="variable" or output="var".
And when using it to specify a custom variable you could just specify it in that parameter using something like output="variable_myStores", output="variable:myStores", output="variable@myStores", or output="myStores.var".
And then to simplify things even further, instead of usingtemplate="/before-after/list.tpl" for outputting to a custom template, then instead use output="/before-after/list.tpl"to indicate the output is handled by the custom template.
If you go this route, the options for the output parameter would then become:
variable | yourcustomvariable.var | yourcustomtemplate.tpl
Basically, you're no longer specifying a template, you are specifying how the output is being handled which can be by use of a variable, custom variable, or template. And that would leave you using just one parameter to handle all cases so there is no confusion about using both.