So, the specific element that makes the Mustaine pickups active is that the preamp is actually in the pickup housing instead of being separate?

This guitar specifies that the pickups are passive but that the separate preamp is active.
Last edited by bachfantasia at Jan 13, 2015,
Passive means no external power source is used.
Active means an external power source is used.
That's it! Doesn't matter if it's a pickup or a preamp or anything else.

So you can have passive pickups running into an active preamp, but the minute you build a preamp into the pickups such that you have to have a battery connected to use the pickups, they're active. The presence of a power source (for amplification) is all that matters in the definition. The Mustaines won't work without a battery, so they're active. The Lukather set work just fine without the active preamp (preamps are basically active by definition) and require no power of their own so they are passive.
Some kind of marketing hoopla, the otehr guitarist in my band dropped Mustaine pickups in his guitar and the installation for it was the same as your garden variety EMGs. They're just trying to make them more "distinct", thus "special", thus more $$$.