They mean differently.
$\color{green}{\Large\bullet}$ Absolute value of $x = |x|$ and is equal to $x$ if $x \geq 0$ or is equal to $-x$ if $x < 0$.
$\color{green}{\Large\bullet}$ Modulo, usually refers to the type of arithmetic called modulo arithmetic.
For example, because $13 = 4\times 3 + 1$, we write $13\ \equiv\ 1\ (\textrm{mod}\ 3)$. In common mathematical language, it is taken as "$13$ is congruent to $1$ modulo $3$".
$\color{green}{\Large\bullet}$ Modulus refers to the magnitude/length of a vector.
Added
How about “An introduction to the theory of Numbers – by Niven Zuckerman” and “Pure Mathematics I & II by F. Gerrish”?
Those names in question have been commonly used by others and sometimes even interchangeably. But, in the books mentioned above, they are clearly and distinctly defined.
The only confusion comes from the “$|…|$” sign, which has been used both for the absolute value of a number and also as the modulus of a vector. Therefore, some used the “$|| … ||$” for the latter to make the meaning distinct. Some don’t even bother when the context is clear or when the readers should be able to distinguish their difference.
Simple answer:
Indeed, if you only consider autonomous differential equations, the concept of (local) flow has nothing to add, although as always being an additional point of view it helps in understanding or perhaps even finding properties that otherwise could be missed.
Not so simple answer:
However, it also happens that the concept of flow is much more general and need not be related to a differential equation. It can be associated for example to a stochastic differential equation, a delay equation, a partial differential equation, or even be associated to multidimensional time, etc, etc.
Complicated but more complete answer:
Having said this it may seem that the concept of flow is something more general than the set of solutions of a differential equation. This is also not a good perspective, since there are generalizations of an autonomous differential equation, even general nonautonomous differential equations, that don't lead to obvious concepts of flows.
The trick of adding $t'=1$ is clearly unsatisfactory in many situations (such as when compactness is crucial), leading for example to the study of convex hulls or lifts in the context of ergodic theory (but leading always to infinite-dimensional systems).
Best Answer
Short answer: I don't think you should worry, as long as you understand this sentence, which says that a function is the whole rule, not a result of applying a rule.
tl;dr
In the first quoted sentence each "value" is a named object, $x_0$ in the domain and $f(x_0)$ in the codomain.
The problem with "values" comes up because functions are often described using a formula with a "variable", usually $x$. The point of the discussion is to make clear that when the function is defined that way, as in "$f(x) = x^2$" , there is really "no $x$" in the definition,
Here you say "value of $f$" because "value $f$" makes no sense. The modifier "value" belongs before a number.
Then this last one is really tricky. Here you say "value of $x$" because you are thinking of $x$ not as a number but as the identity function on the domain.