The second FTC can be confusing with all those letters x and t floating around. Chillax. We're going to straighten them out here.
A function and its derivative use the same independent variable. If we have a function f(x) and we take its derivative, we get a function f '(x). We don't get f '(t).
Working backwards, a function and its antiderivative also use the same independent variable. If we're trying to build an antiderivative for f(x), the antiderivative must also be a function of x. So we need to end up with an antiderivative F(x), not F(t). This is why we use x as a limit of integration: when you change the upper limit of integration, you change the value of the definite integral.
When x is a limit of integration, we don't want to use it as the variable of integration too. Writing
is horribly confusing. Other than that, it doesn't matter what letter we use for the variable of integration. We could write
When we change the letter, all we're doing is relabeling the horizontal axis. Regardless of the labeling, we're finding the weighted area between f and the horizontal axis from a to x.