I am going to rule out bear types one by one. But let’s first include a few assumptions and identify special cases.

Digging a pit, or a “trap”, as it is called in some versions of this question found on the internet already puts some limits on the situation. This really can’t be occurring at a zoo, because none would create such an environment for the bear to fall into. Nor any sort of privately owned animal. We have to be talking about wild animals if we have a creature falling into a pit, especially if it was set as a trap.

While air resistance and being above sea level (thus experiencing a change in the local acceleration of gravity) might be a factor, these seem to be special cases we can deal with after considering sea level. So let’s look at that first.

The math using the above numbers says that the acceleration of gravity at the place the bear fell, considered to be sea level for this calculation, is 9.8085 m/s^2.

Looking up on the Smithsonian's Physical Tables (Smithsonian Physical Tables) I see that the latitude for the location of this well is approximately 47º 30' N or S. So knowing this, we can start ruling out bears by their naturally occuring habitate and range.

So unless we are including zoos, the bear is not white. Polar Bears don't range that far south. And zoos really don't have wells, so that should be safe. (Also, the original question actually uses a trap, not a well. Zoos definitely don't have pit-traps.)

• There are no bears in the southern hemisphere, except Sun Bears and the Speckled Bear, but none of those south of 20º S. So we now know we only need to consider 47º 30' N.
• Polar Bears do not range that far south, so we can rule them out. (Note, below I will consider them for wind resistence during the fall and altitude above sea level).
• The Syrian Brown Bear lives too far south. We can rule this bear out.
• The Sloth Bear lives in India. Too far south. We can rule this bear out.
• Giant Pandas have a tiny habitat. Its northern extent is about 35ºN, so again too far south (and here again, very important that zoos are out of consideration. About 15% of the world population of Giant Pandas live in captivity.) We can rule this bear out, since we are ignoring zoos.
• The Tibetian Blue Bear, never before photographed, would be great to include, but again, too far south.
• The Asiatic Black Bear has a range from about 40ºN to 50ºN, so it is a contender.
• The Brown Bear (also called the Grizzly Bear and the Eurasian Brown Bear) has its core range in Russia, Canada and Alaska (north of 50ºN), but does have a low-population range in the Rockies that extends as far as Yellowstone-Teton in Wyoming, which is about 44ºN. Also found in Eastern Europe, as far south as Greece. And in Central China. Also a contender. This bear is a contender.
• Note - In the original test I found, the Grizzly was listed separately on the bullet item list as a "gray" colored bear. See” Creating Difficult Questions.
• The American Black Bear extends from Mexico to the Arctic Circle. Very much a contender.

More about these creatures here: Bear on Wikipedia

So that really just leaves us with a black-colored bear or a brown-colored bear (and the "gray-colored" bear from the original test; the grizzly).

The other sets of answers I have found on the internet discuss the fact that this was a trap, made to catch bears on purpose. In China, only the black bear has monetary value, praised for its claws and (illegal) bear bile used in Chinese medicine. Similarly the brown bear lives in mountainous regions almost universally and trying to dig a 19m deep pit in a mountain region is nearly impossible. (The same is probably true of a well big enough for a bear to fall into.)

So I am going to also go with Black Bear.

Corner Cases and other Considerations

—Altitude—

Comments have been asking about altitude. Let's take a look. To change the acceleration of g at 50º to include a possible polar bear on the shores of Hudson Bay, what altitude do we need to be at?

We go from 9.8085 to 9.81053.

So solving for r2-r1 (altitude above sea level), we get: 659m (2162 ft). Which would be very significant; if Hudson Bay wasn't at sea level. Polar bears are coastal creatures. There is no way to use high latitude mountains to change which bears are contenders in the above list.

—Air Resistance—

People have also been asking about air resistance. A 450 kg polar bear might be falling slower, thus appear to be at a lower latitude when calculating the acceleration of gravity. To get a change of g from 9.81053 (see above) to the measured value in the question, we need a mere .909N of force acting on the bear.

The maximum speed of the bear is easy to calculate. It's 19.617 m/s.
So, what sort of air resistance can we estimate?

F is the drag force (average of .909N)
p is the density of air (1.225 kg/m^3)
C is a coefficient for drag
A is the cross section of the bear (about 1.5 m^2)

So force is proportional to the square of velocity. While we do get up to about 43 mph, we are also resisting a 450 kg weight with air resistance, which isn't mass-relative. Someone can do the differential equations needed (Falling Body with Air Resistance), but charts show that this force does not become significant until you get close to terminal velocity. I feel we can ignore it for a 2 second fall.