The use of the term "mini humbucker" is somewhat general, and is usually used today to refer to a basic form of pickup, rather than to the specific construction of the pickup that fits that form. The term today usually is meant to cover *both* Gibson-type mini humbuckers: the Firebird pickup and the New York mini humbucker pickup.
The general mini humbucker form is originally from the pre-Gibson-era Epiphone company. It started out as a single coil, referred to (not sure whether officially or not) as the New York pickup.
In the late '50's Epi was bought by Gibson's parent company, and the Gibson and Epi companies soon started sharing technology and designs, as well as production facilities. In the early '60's, this basic Epi single coil form was taken and developed into a humbucker, in both bass and guitar versions. It was used in various Epis and some Gibsons (like their two-pickup basses). I call these New York mini humbuckers in order to keep things straight, though I don't know the official brand terminology for them. They eventually found their way into Les Paul Deluxes when the LP was reissued by Gibson in the late '60's. The design of these pickups is very similar to a full-sized humbucker – just smaller. The main difference is that there are no individual slugs on the mini – just a single large "blade" slug, which is magnetized by the bar magnet that sits horizontally below the coils (just like a regular sized humbucker).
A few years after Gibson developed these New York mini humbuckers, a different design of pickup, using the same basic form, was developed specifically for Gibson Firebirds. These were a very different design. They used a "blade" permanent magnet inside each coil, a steel plate bridging the blades (sometimes on top of the blades and sometimes on the bottom), and were covered with a smooth cover (no adjustable poles). I call these Firebird pickups.
As far as Duncan offerings go, the standard production pickups that they call Vintage Mini Humbuckers are built like Firebird pickups, but with "S" engraved on the covers. The Seymourized versions might be overwound versions of the same basic construction too (though I remember being told by Jeremy that the Custom versions are a bit different). In order to get a New York mini humbucker from Duncan, you have to go to the Antiquity line (where you can also find Antiquity Firebird pickups, which are basically Vintage Mini Humbuckers that have been equipped with non-engraved covers, and "Dun-Aged").
So, bottom line, when someone says, "Put in a mini humbucker," it doesn't just mean one thing. You have to pick Firebird style or New York mini style...and the tonal differences between the two are actually quite plain, so don't think of them as the same thing just because they'll fit the same pickup rout on your guitar.
