1.
The second paragraph in U.N.'s post seems to make the same point as your point about using just
$C$.
2.
I guess something like this is meant about the independence:
Problem: Find a function
$f \colon {\Bbb R}\backslash \{0\} \rightarrow {\Bbb R}$ such that
$f'(x) = 1/x$,
$f(-1) = 1$, and
$f(1) = 2$.
Solution:
$f(x) = \cases{\log(-x) + 1 & $x <0$ \cr \log(x) + 2 &
$x>0$ \cr}$
This function is not of the form
$\log |x| + C$. And the general problem with conditions
$f(-1) = C_1$ and
$f(1) = C_2$ has a solution space of dimension 2.
3.
Another way to look at the dimension of the solution space is this:
The general real antiderivative given by a particular solution
$\log|x|$ plus any solution to
$Df=0$ over the domain of
$1/x$.
The condition
$Df=0$ implies only that
$f$ is locally constant. If the domain consists of a certain number of disjoint open intervals, then the dimension of the solution space of
$Df=0$ will be equal to the number of intervals. In the case of
$1/x$, in which the domain consists of two intervals, the dimension is two, which may be parametrized by two independent parameters
$C_1$,
$C_2$.
More generally, for a rational function
$p(x)/q(x)$, if
$q$ has
$n$ real roots (counted without multiplicities), then the space of its antiderivatives will have dimension
$n+1$. However, its complex domain consists of a single connected component, and the solution space consequently can be parametrized by a single constant.
4.
The domains in which locally constant functions play a role tend to be in higher mathematics, not in first-year calculus (or even in most second-year differential equation courses).
I think that's the point of the paragraph that begins "In practical work..." I referred to in point 1 above. In most problems in introductory calculus, the domain can be restricted to a single interval and a single constant
$C$ parameterizes the solution space.
The problem in point 2 above, while completely elementary, has not appeared in any calculus book I've read or taught from, as far as I can recall. Clearly it is not considered an important point to explain in introductory calculus.
On the other hand, locally constant functions arise in some branches of engineering and other fields in which you have continuous inputs and discrete outputs. DSolve[] and NDSolve[] can handle locally constant functions with DiscreteVariables and WhenEvent in some situations. I don't think it can handle this case though.
