Are the integers a, b, and c consecutive?
(1) b – a = c – b.
(2) The average of a, b and c is equal to b.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) The first statement simplifies to:
2b = c + a, so b = (c + a)/2.
This means that b is the average of a and c, but does not imply that the three integers are consecutive because we don't know how they're distributed. All this tells us is that the difference between a and b is the same as the difference between c and b. For example, they could be 4, 5, and 6, or they could be 3, 5, and 7. They could even be 5, 5, and 5. So Statement (1) is not sufficient.

Statement (2) gives us the same information: we know that the average of the three integers is b, but we still don't know the distribution of those integers. Any of the previous examples would have an average that is equal to b. So Statement (2) is not sufficient either.

Combined, the statements are insufficient because we still don't know how the integers are distributed. In fact, any time the statements tell us exactly the same thing, combining them will not help. Therefore, the correct answer is choice (E).
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I know that x, x + 1, x + 2 are consecutive, but are x, x + 2, x + 4 consecutive as well by definition?

The term "consecutive" depends on the set of numbers we are considering. If nothing else is specified then it defines consecutive integers.
Three consecutive integers will be x, x + 1, x + 2, where x is an integer.

In many cases the set of numbers we are considering is specified.
For example "consecutive even integers" defines that we are considering consecutive numbers in the set of (among) even integers.
{..., -4, -2, 0, 2, 4, 6, 8, ...}
Three consecutive even integers will be x, x + 2, x + 4, where x is an even integer.

"Consecutive prime numbers" defines that we are considering consecutive numbers in the set of (among) prime numbers.
{2, 3, 5, 7, 11, 13, ...}
Three consecutive prime numbers are 2, 3, 5; or 5, 7, 11 etc.

There can be many other specific sets with term consecutive: "consecutive multiples of 5", "consecutive months", etc.