To avoid the confusing (to me) notation, let's call $g(x)$ the antiderivative of $f(x)$, such that $g'(x) = f(x)$. I've seen $F(c)$ defined as the definite integral of $f(x)$ from $0$ to $c$ with respect to $x$ in some situations, and as $g(x)$ in other situations. While these definitions are equivalent if $g(0) = 0$, they are different if $g(0)$ is not $0$ since the former definition is equivalent to $g(c) - g(0)$, while the latter is equivalent to $g(c)$. Assume that $f(b)$ for some value $b$ is known, so there is only one correct $g(c)$ (as opposed to infinite antiderivatives) and thus I am able to compute $g(c)$.
Without an explicit definition, does $F(c)$ take on the value of $g(c) - g(0)$ (the former definition) or $g(c)$ (the latter definition). In other words, is $F(c)$ the definite integral from $0$ to $c$ of $f(x)$ with respect to $x$, or the one indefinite integral of $f(x)$ evaluated at $c$?