Print button action - prints the PDF format

Can the print button on the form use the PDF template instead of printing what the form looks like? 

Unfortunately, there is a separate print button for the PDF template and for the print button on the form. If you want, rather than allowing the user to print their submissions or forms, you can include PDF reports in the email template that will be sent to them once the form is submitted.

Guide: How-to-include-submissions-as-PDF-attachment-in-the-email-notifications

Ideally they would print the form but then it won't include a submission ID or other identifier.  Is there a way to get a printout instead of printing an emailed PDF?

Submission-ID and date of submission are generated once the form is submitted so I do not think it is possible. If you like I can make a feature request with your requirements, but we cannot guarantee that the feature will be implemented or cannot provide any time frame for implementation at the moment.