Overriding menu.html.twig just for the primary menu
Answering my own question here, this will probably be useful for those looking to debug twig. 


Activate devel Kint (part of the Devel module) on the modules page or with Drush
In your site /sites/default/services.yml change the following settings:

parameters:
  twig.config:
    debug: true
cache: false
auto_reload: true
Then in the template you want to debug, place this code in it:

{{ kint() }}


Now you get a nice tree of variables that prints out similar to what you would get if you did an old style kpr(get_defined_vars()); in Drupal 7. 




I see from the debug printout that the name of the menu is simply main.
Now I change the overridden template in my custom theme's template folder to menu--main.html.twig
Voila, the template is only specific for the primary main menu now
Customize as a themer to your hearts content.
Throughout this process, don't forget to clear cache and change / remove any debug code when you are done.


In addition to this, with debugging active, Chrome prints out possible template suggestions a la devel_themer so you don't necessarily have to use Kint.