I looked at the code a little bit. Hopefully I can clear some things up.
The player history message for nuking is added by the thermonuclear bomb item; it would be simple to add a similar message to the Trigun. But, as you found out, the majority of nuke-related stuff including the Fallout counter are done for any source of nuking.
The default condition for completing a level is to exit the level with no enemies left alive (which is a huge pain on Nightmare). A few levels (like Hell's Arena) have a special condition instead (like requiring that all the waves are complete on Hell's Arena). The Vaults doesn't have any special condition, and I don't recall any of the special conditions mentioning nukes. This means that nuking levels will count for clearing special levels and the associated general medals (Conquerer, Thomas's).
The level-specific awards use entirely separate code however. For the Vaults, Gold+ badges will not be awarded if the level has been nuked. The same goes for Hellgate, Unholy Cathedral, and The Mortuary.