FlatWoodDrill BitSet

But I've also seen people use the term for bits where the cutting edge is just angled inward, so that the outside edge cuts first, like this one:

Both styles do a better job in wood than normal twist bits that are often ground to an angle of 135° because brad point bits don't wander, don't need a pilot hole, and cut a cleaner edge.

Flat Drill BitSet

With regard to the difference between a wood drill bit and a metal drill bit, the respondent talks about a wood drill bit having a "spar/brad" and a "lip," but doesn't point to (or show such) in the accompanying photos!

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Flat Drill Bitfor metal

As you've quoted, the spar/brad is the point at the center of the bit. One can find spade bits and twist bits with a brad point.

As for how to tell them [wood bit vs. metal bit] apart -- wood drill bits have two distinguishing features: a spar (also brad) and a lip. The spar helps in keeping the bit centered when starting the hole and prevents wandering of the bit on the surface of the wood. The lip acts like a chisel, slicing through the wood fibers in the circumference of the drilled hole, improving the quality of the finish. This is most important for through holes, where the lips helps reduce "blow-out" of the wood when exiting the wood on the other side. Wood bits are typical black and silvery in color, but other colors exist.

FlatWoodDrill Bit35mm

The lip being referenced as being used to cut the circumference of the hole means that there's a pair of tips extending deeper into the surface than the "scraping" portion of the spade bit.

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Wideflatheaddrill bit

Forstner bits will also have spar/brad centers and lips but will occasionally (usually) have a rotating chisel like surface that extends from the center to the perimeter.

Metal drill bits, on the other hand, have just an angle at the tip. They often are fully black (plain HSS), golden (often titanium oxide finish) or coppery (often cobalt steel) in color.

The "brad" on a brad-point bit is the center point. It bites into the wood easily and helps to keep the bit from wandering when you start your hole. The "lip" is the outer edge of the bit, which protrudes a bit to slice the wood fibers at the edge of the hole so that you end up with a very clean hole. Personally, I think of lipped bits as having an outside cutter that's distinct from the rest of the cutting edge, as in this one from Lee Valley: