Is a bad bishop a bishop that is trapped by your own pawns? (For example, all your pawns are on dark squares, so the dark squared bishop is bad.) Or is it when your opponent's pawns trap your bishop's mobility? (For example, they control the dark squares which your bishop controls, so the bishop is bad).
What you defined is an "inactive bishop". An inactive bishop can be a bad bishop, but they are technically two different situations.
A "bad bishop" is a bishop that is on the same color as it's own center pawns. An "inactive bishop" is one that is blocked by it's own pawns.
When evaluating a bishop, there are two totally separate things that must be considered. The first is whether a bishop is "good" or "bad". These are very much misnomers because they have nothing to do with the worth of the bishop. The second consideration is whether the bishop is "active" or "inactive". Finally, there may be mitigating factors as well (is the bishop doing something useful?).
Good vs Bad
A good bishop is on the opposite color as its pawns. A bad bishop is on the same color as its pawns. For example if the white pawn structure is c3, d4, e3, f4, then the light squared bishop is the good bishop and the dark squared bishop is the bad bishop.
Active vs Inactive
An active bishop is outside of the friendly pawn chain (in the above example, if the bishop is on e5, then it is clearly both active and bad). An inactive bishop is trapped by pawn chains - either the player's color or the opponent's.
There are two main mitigating factors. The first can be remembered by a common phrase: "bad bishops defend good pawns". This means that a bishop might be very bad (same color as all the friendly pawns) but if it is defending them from attack in the endgame, then it might be enough to hold a draw. Usually it is much harder to win with the bad bishop though.
Several useful answers were given here last year, answers which illuminate the exceptions to the rule so well that—when the answers are read together—they seem almost to leave the rule itself in shadow. Let me cast my flickering light therefore directly upon the rule.
The good and the bad bishop
The technical adjectives "good" and "bad" have much to do with the worth of a bishop. A good bishop very often wins a game of chess. Indeed, unless one has a material lead of two pawns or better, has a passed pawn, or can push or trade pawns to make a bad bishop better, to win an endgame with the bad bishop can be hard. On my own, limited experience, I should go further: I would say that to win with the bad bishop, without a strong counteradvantage to compensate so great a disadvantage, were seldom possible absent a gross enemy blunder.
Good against bad bishops in the endgame
Endgames that feature opposing bishops on squares of the same color are quite common in chess, as you undoubtedly know. As soon as such an endgame seems probable, one might consider pushing as many of one's pawns as possible, on both flanks, as fast as possible, to squares of the color the bishops cannot reach.
In class-D and class-C play, to have pushed pawns in so simple, innocuous-seeming a manner often wins the game 20 or so moves later; insofar as opponents at this level, not grasping the principle, have no idea why you would spend valuable late-midgame time to push seemingly random pawns. One, quiet attraction of winning a game in this manner is that it leaves one's opponent with little notion as to why he has lost. (Indeed, I suspect that it is in large measure because of the mysterious bad-bishop principle that class-D players tend to dislike endgames, why some will launch unsound attacks just to avoid endgames. They repeatedly lose endgames for no visible reason and don't know why. All endgame maneuvering, all endgame calculation seems mysteriously futile to them. The unseen reason, of course, is that they have lost, or at best drawn, before the endgame ever began by letting opponents choose during the midgame the color of the squares on which endgame pawns must stand. They think you and your bishop cleverly nimble, hardly realizing how easy it is to be nimble when one's bishop is good.)
Other endgames involving bishops
Endgames that feature opposing bishops on squares of different colors admittedly work on another principle, though such games are significantly less common in practical play. In bishop-against-knight endgames, which are common enough, however, the square-color principle again emerges, though in a less pure form.
Why the good bishop is good
The good bishop is good for three principal reasons:
The bad bishop does possess the counteradvantage that it and its pawns can guard one another but, except maybe when the pawn is a rook pawn, such a counteradvantage seldom compensates.
Definitions and remarks
Others have already answered your question about definitions well enough, so here I will add only that I have read the definition as given both with and without reference specifically to central pawns. When speaking of the opening, I should call a bishop technically bad when central pawns stood on squares of its color, without much reference to flank pawns, as others have noted. However, from the early midgame, my view should gradually broaden toward the flanks, eventually taking all pawns into account of the bad bishop. The two definitions are not unrelated, of course, for the technically bad bishop of the opening is more likely to become the dispositively bad bishop of the endgame than is a technically good bishop, unless its owner either has taken steps to improve it or has swapped it off.
Consider establishing a good bishop when you can. It may win you some games otherwise drawn or lost.
The term itself is rather simple. A "Bad Bishop" is one that is on the same color as most/all ( the higher the percentage, the worse the bishop) and is behind them, that is to say, the pawns are preventing it from attacking the enemy.
If you can get your light-square bishop on the outside of your pawns, it's will no longer be bad, though it may have reduced mobility.
Defensive or not, a bad bishop is bad (the logical end of defensive warfare is surrender - Napoleon) and its bad because it does nothing to limit, or otherwise influence, the actions of the enemy.
Never willingly accept a bad bishop unless you get something in return.
A "bad" bishop is one that is not performing the normal function of a bishop. That is usually one behind its own pawns. The test is, "if I replaced the bishop with a pawn, would it make any difference?" If the answer is no, then it is a bad bishop.
A Bishop can be behind a wall of pawns and still be very useful. Say there is a bishop on g7 in front of its king on g8, and behind pawns on h6, g5, and f6. As the "linchpin" of the defensive system, the bishop is very useful, even though it has little mobility. (Unless this is a situation where all the action is on the queenside, in which case the bishop would really be "bad.")
Every piece has a strength that depends on the position on the board. More specifically, it depends on the pawn structure and piece placement. As you correctly point out, a bishop trapped behind its own pawns (e.g. white has Bc1, pawns on b4, c3 and d2; black has pawns on b5, c4 and d3) is a bad bishop. In general, a bishop that has almost no available squares and/or no targets to attack is a bad bishop. For example, the rule of placing your pawns on squares that are of the opposite color to the squares that the bishop controls is related to both of these factors. If you have a dark squared bishop and you place your pawns on light squares, two things occur.
Another example of a bad bishop is a bishop that has to support its own pawn chain, e.g. white has Bb3 and pawns on c2 and a4. As soon as the bishop moves, both pawns become unprotected. A bishop needs
The less these conditions hold true, the worse bishop you get!