Covert means shoot men in the face: WinBack (PS2)

WinBack: Covert Operations on the PS2 is an update of a Nintendo 64 game of the same name- a third-person cover-based shooter, but one from 1999. It improves the graphics and adds mediocre Japanese voice acting and awful English voice acting.

It's apparently set in a world where the ability to jump is nonexistent. At least, that's what the architecture and traps imply.

It's not a horror game, but the black guy dies first. Like a chump.

Your main character is named Jean-Luc Cougar. Another teammate is named Keith Birdy. There are (sadly) no animal puns in this game.