When you create a product from the menu, what you inmediatelly see is the product template.
If you go to the tab "variant", you will see all products variant related to that product (variants = product.product, yes product.product is a children of product.template).
By default when you create a product.template, a default variant with some of the template values is created.

John, you are right that product.product is a child of product.template.  Changes in product.template fields does propagate to its children (all products that has the same template).  Changes to product.product does not propagate to product.template.

You inherit product.template if you want the property you are adding to affect a group of product variants (product.product).  You inherit the product.product if you want the property that you are adding to affect only one product variants (meaning each product variant can have different value).

Note that I did not put changing in the above paragraph because once a field is defined in product.template, it better stays there (and vv).