|1) The Matrox card is a capture card that comes with drivers to integrate it with Premiere;|
2) Without third-party drivers the only capture card Premiere recognizes is an ordinary OHCI Firewire card;
3) Neither are like the ATI card you referenced, which is technically a 'computer graphics adapter' ... though commonly referred to as a 'video card' ...
4) The analog outputs of the Matrox card are the correct way to feed an attached TV monitor, though many make do with routing the video through the Firewire to an attached DV device that is in turn attached to a TV monitor;
5) The Matrox (and maybe the Canopus) video editing cards are specialized feature-enriched accelerators for Premiere.