In my opinion, the 'confusion' lies in the fact that the integral sign (∫) is used in two very different ways.
For many people, this leads to the conclusion that there are two 'types' of integrals, which is not right.
The actual 'integral' is a definite integral. Referring to the (relatively easy to understand) Riemann-integral, it is clearly defined in such a way that it returns what our understanding of an area is. The Riemann sum represents an approximation of the area under f(x) which we're integrating, the limit (the integral) is by definition the area.
Now, when dropping the limits of integration, we get what's been given the name indefinite integral. It should be clear that this isn't another type of integral, it is merely (a perhaps unfortunately chosen) notation for a concept we can perfectly define without 'integration theory'. Given a function f(x) ([a,b] to R), we call a differentiable function F(x) ([a,b] to R) a primitive function of f(x), if F(x)' = f(x), for all x in [a,b].
Because of historical reasons, we introduced the notation:
With F(x) defined as above. But we can also 'define' the indefinite integral, as a definite integral:
Here, f is still a function of x, but we use the dummy variable t to not mix integration variable and integration limit. The (-)F(a) plays the role of the constant of integration in the notation before, cfr. the fundamental theorem of calculus, of course.
Now, since (*) may hold for more than one function (trivially: you can always add a constant), we indeed don't have "a primitive function", but a "set of primitive functions", which is exactly what we mean with the notation of the indefinite integral.