The distinction you make is correct in the sense there is indeed a fundamental difference between proving P by assuming not-P and reaching a contradiction and on the other hand proving not-P by assuming P and reaching a contradiction. However, both are called proof by contradiction. It is just plain wrong to say that the second kind is not proof by contradiction. It has been called like that for more than two millenia, whereas intuitionism is a 20th century idea. Besides, if you insist on the difference, then you have to distinguish between positive and negative mathematical properties. For instance, in your example, "finite" is positive and "infinite" is not-finite, so negative. For a classical mathematician, which is most of them, this is actually an undesirable distinction that depends on how things are defined, and is not intuitively clear.