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.
Thus, a function $f$ should be distinguished from its value $f(x_0)$
at the value $x_0$ in its domain.
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,
since $f(x)$ and $x^2$ should both be understood as the value of $f$
at $x$.
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.
valid for all real values of $x$
A term is a "name": variables and constants are terms.
In addition, "complex" terms can be manufactured using function symbols.
Example: $n$ is a variable, $0$ is a constant and $+$ is a (binary) function symbol.
Thus, $n,0$ and $n+0$ are terms.
Formulas are statements.
Atomic formulas are the basic building blocks for manufacturing statements.
They are formulas that have no sub-parts that are formulas.
They are manufactured using predicate symbols, like e.g. $\text {Even}(x)$, equality and terms.
Thus, $\text {Even}(n), 0=0$ and $n+0=n$ are atomic formulas.
With connectives and quantifiers we can write more complex formulas, like: $\forall n (n+0=n)$ and $0=0 \to \forall n (n+0=n)$.
Expression can be a "generic" category: it may mean a string of symbols.
We may call well-formed expression a string of symbols that satisfies the rule of the syntax.
If so, it is either a term or a formula.
Best Answer
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).