For what it’s worth I’ve checked the database and Contact form 2 content is apparently all there, so I figured that, based on the other shortcode I had deduced correctly (in order to have Contact form 1 appear in the page text) that the short code for the duplicated form would be
[contact-form 2 "Contact form 2"]
This short code doesn’t get the form to appear, I get this instead
[contact-form 404 "Not Found"]
Bit frustrating.