It seems the underlying difficulty is that due to the coincidence of some
symbols used in the two parts of the Fundamental Theorem of Calculus
in your textbook, you are linking things together that are not
really meant to be linked in that way.
The explanation of the FTC on the MathWorld site
does not even have two "parts"--it presents two separate theorems.
The "first" theorem says:
If $f$ is continuous on the closed interval $[a,b]$ and $F$ is the indefinite integral of $f$ on $[a,b],$ then
The "second" theorem (according to MathWorld) says (paraphrasing slightly) that
If $f$ is a continuous function on an open interval $I$ and $a$ is any point in $I$, and if $F$ is defined by
at each point in I.
You have not quoted your textbook verbatim, but from the sequence of
formulas in your question it would appear that MathWorld's
"second" theorem is the first "part" of your book's theorem,
and vice versa.
This is all fine, because the two theorems (or two parts of "the" theorem)
are logically independent of each other--neither one assumes the other
as part of its statement.
In particular, all the names of functions and variables in either part of
"the" theorem are tied only to functions and variables of the same names
in the same part of "the" theorem.
We could just as well write:
If $g$ is continuous on the closed interval $[c,d]$ and $G$ is the indefinite integral of $g$ on $[c,d],$ then
while leaving all the letters $f$, $I$, $a$, $F$, $x$, and $t$
unchanged in the other part of "the" theorem.
You don't have to assume the two parts are talking about the same
functions and numbers at all.
To put it another way, the two parts of "the" FTC are doing two
One part of "the" FTC tells you how to compute
definite integrals of a function if you happen to know
an indefinite integral (antiderivative) of the same function.
The other part constructs an antiderivative of a function
from a particular definite integral of the same function.