To make sure we can give you the proper information, you need to post links to the actual drill bit sets, or list the item number, otherwise it's mostly speculation.
First, to answer your question about "Bronze Oxide" finished drill bits. The Bronze Oxide finish on drill bits has nothing to do with bronze, except for the color. The finish is used on some drill bits including regular High Speed Steel/ HSS types as well as more specialized drill bits such as ones manufactured out of Cobalt steel. Supposedly Bronze Oxide helps with "Lubricity" although how much probably depends on the material you'r drilling, and the lubricant you'r using if any.
Black Oxide is the finish found on most general purpose drill bits. The Black Oxide provides protection against corrosion, and I believe it also helps hold drilling lubricant. It's generally good for drilling steel and some iron alloys. I believe Black Oxide isn't recommended for drilling aluminum because it can cause problems with galling. If you have to use the drill bits for drilling wood occasionally I would also recommend against Black Oxide finished bits. I'm not sure exactly why, but the bits don't seem to clear wood chips as well.
Bright Finish drill bits, are bits that have been ground but not given a surface finish. Usually bright finish bits are recommended for drilling aluminum, and possibly some brass, bronze, or copper alloys. I can't remember exactly of the top of my head. Titanium Nitrate coated bits, are usually preferred over the Bright Finish bits for drilling aluminum, but good Titanium finish industrial quality bits are very expensive, as opposed to the cheap ones that are found all over the place. Bright finish bits are also probably best for general purpose drilling in materials like plastics and wood.
Some Triumph drill bits, such as their regular Thunderbit brand ones, are finished with Black Oxide in the flutes, and Bronze Oxide on the outer surface. The Bronze Oxide provides Lubricity to the outside of the drill bit, and the Black Oxide hods lubricant for better chip clearing, and cutting performance. This is the finish found on the Standard Thunderbit sets Snap-On sells.