But, I'd like only the Primary domain mapping to work.
When you add the domain map there is a radio that allows you to select the primary domain for that site. Use the radio to select the mapped domain as the primary. The other URLS will redirect to the primary.
The 5 checkboxes in the settings (when checked)
Remote Login - logged in user is logged into both domain.com/site and mapped-domain.com
Permanent redirect - a permanent redirect header is used (vs a temporary redirect) when redirecting from non-primary to primary
User domain mapping page - the domain mapping tool screen is shown on sub sites under tools
Redirect administration pages to site's original domain - site dashboard runs under the domain.com/site/wp-admin URL
Disable primary domain check - disables the redirect from non-primary domain to primary domain