I'd say that's just the flavor of the Thaumaturgy for that kind of magic.

The idea is to model being able to better function while damaged than most people. A character with this stunt shouldn't be getting hurt any less--they should just be able to keep going longer while injured.

From a non-mechanical point of view, keeping going longer while injured reads to me as they can take more hits before being taken out.  Does this sound about right?

Mechanically, being taken out is when you have to fill in a space on your stress track and have no space to fill in.  There are lots of different ways to allow a character to get hit more before being taken out.

• Have more stress boxes to fill in.
• Have more consequences to spend to reduce incoming stress
• Take less stress per hit

Both the first 2 allow one to cast bigger, stronger spells, by using the higher stress boxes and/or consequences, which you've said you don't want (and doesn't fit the fluff of ignoring pain).  As for the third, the easiest way to take less stress per hit, mechanically,  is Armour. 

Yes, most of the time Armor (the power/ability) is used in the book as just that, armor (scales, flak jacket, etc).  But there are lots of other things that can be used for something that isn't exactly what the name implies. The one that jumps to mind is Breath Weapon.  I'm sure not every creature/character that has the power called "Breath Weapon" is actually breathing something on their target.  Some are just attacking at range (which is mechanically what Breath Weapon is doing).  Mechanically what Armor is doing is allowing you to take more hits before being taken out.  Sometimes that's because each hit is doing less damage (in a non-mechanical sense) to you.  But there's nothing that says it can't mean you just ignore the pain (or some of the pain) of the hit, so you won't go down as easily as someone who doesn't have it.

Speaking of Special Investigations, I always liked the covert gov'nt military group from Buffy the Vampire Slayer...I forget what they were called.  They'd hunt supers and take them down to do experiments on them.  All the soldiers were Jacked up on military "enhancement" drugs that gave them strength and speed and fast healing.  THeir base of opperations was in secret tunnels under the university.

Edit: they were called The Initiative

We have a faction exactly like them in our current game, but they are a private corporation instead of government funded.  They capture and experiment on supernaturals to try and duplicate their powers.