Unfortunately, parts of the section about integration are wrong and contribute to some of the widespread confusion about the FTC, which often seems reduced to a mere tautology (which it's not; it's a deep result).

First, the "indefinite integral" is really just the antiderivative (and as such, "the" indefinite integral is only unique up to a constant). This doesn't follow from the 1. FTC, but purely by definition.

OTOH, the definite integral is not "defined" as being the difference of antiderivatives; it's defined in terms of Riemann sums (at least in elementary calculus), as explained further below.

What the two parts of the FTC do is proving that those two notions, which have no a priori reason to be related, are in fact related in a particular way.

The first part of the theorem says that, if f is "nice" (in particular: continuous), the antiderivative exists and can be expressed through the definite integral with a variable upper bound.

The second part says that, if f has an antiderivative, the definite integral can be computed using that antiderivative.

It's important to keep this distinction because there are e.g. functions that are integrable but don't have an antiderivative (e.g. a function with a "jump"). In such a case, the FTC tells you nothing and you have to go back to Riemann sums to compute the definite integral.