I can see several different ways to hold a knife and I know how their dynamics are different, but I don't know what each one's called (or why; from what I've seen, the "saber grip" seems to be a way I'd NEVER want to hold a saber).
Saber grip: handle held across the palm, as opposed to straight up and down, with thumb along either the spine or side of blade.
Hammer grip/natural grip: handle held straight up and down in the palm, thumb curled around the handle, looks akin to making a fist.
Reverse grip/icepick grip: similar to hammer grip, but with blade pointing down; a variation on the theme has you cap the pommel with your thumb.
As for why a "saber grip" is called a "saber grip", that seems to stem from several types of cavalry sabers which were designed for giving point and had a dished out indentation on the back of the handle for your thumb. This made it easier to bring the weapon "on line" with it's target.
You seldom use this grip with a saber when fighting on foot, except to thrust, depending on handle configuration.
Since I mentioned it in another post, I'll add half-saber. This is where the knife is held in a saber grip but rotated 90 degrees so the blade is horizontal and the meat of the thumb rests flat on the side of the handle or the ricasso. There's another name for this (some fellow's name) but I can't recall it. The only use of this grip that I know is in surprise assaults - the knife comes from an unexpected angle (held low at the side) and the horizontal blade passes between the ribs. Not a useful grip and not PC either, but I'm throwing it in for definition's sake.
Im not familiar with any actual tactical knife fighting grips but can speak to fencing grips having been on the fencing team in school. Snicks description is really for the foil where you score on thrusts to abdomen area only. A saber, which happened to be my weapon of choice, is properly held with the grip resting in the crease of your second knuckles and your thumb is positioned on the back for leverage. Points are usually scored with slashes and not thrusts.