Here is my interpretation.
The word solution is used in the following context. Find the solution to $$f(x) = b \tag{$\star$},$$ where $f: A \mapsto B$, i.e., find the set of all $x \in A$ such that $f(x) = b$.
The zeros of the function $f$ is the set of all $a \in A$ such that $f(a) = 0_B$, where $0_B$ is the zero element in $B$. Hence, zeros are used specifically in the context when the $b$ in $(\star)$ is $0_B$. To see it in a slightly different way, the zeros of $g(x)$, where $$g(x) = f(x)-b$$ are the solutions to the equation $f(x) = b$.
The term roots are typically used to describe the zeros of a function, when the function $f(x)$ is of the following form: $f: R \mapsto R$, where $R$ is a ring. I believe, the usage was due to the fact that we talk about finding the square roots, cube roots, etc, which was then extended to polynomials and thereby extended to rings.
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
"I've been reading intros to ODE and the problem of terminology has overwhelmed me.
As far as I understand: 1.n-Parameter family of solutions to ODE is a solution in a form cy1+ky2" No. That is a "two parameter" solution to a (linear) second order equation. An "n-parameter family" to a (linear) nth order equation would be a linear combination of n "fundamental solutions".
"2.singular solution to ODE is a solution which can't be obtained from (1)"
Yes, but such singular solutions appear only in non-linear equations.'
"3.general solution is a n-Parameter family of solutions when singular solutions are absent." Yes, that follows immediately from the definition of "singular solution". If there are no singular solutions, that is, no solutions that are not of that form, then all solutions are of that form!
"4.fundamental solution...?" A "fundamental solution" to a linear, homogeneous, nth order differential equation (at x= a) is a solution to the equation such that y(a)= 1, y'(a)= 0, ..., $y^{(n-1)}(a)= 0$ or such that y(a)= 0, y'(a)= 1, y''(a)= 0 ..., $y^{(n- 1)}(a)= 0$, etc.
"5.fundamental set of solutions...?" A set of fundamental solutions all the way from "y(a)= 1, y'(a)= 0, ..., $y^{(n-1)}= 0$" to "y(a)= 0, y'(a)= 0, ..., $y^{(n-1)}= 1".
"What's the point in talking about "fundamental solution" when we have a general one?" If we already have a general solution then there would be no point at all. But before we have a "general solution" we must know such a thing [b]exists[/b]. Have you read through the basic "existence and uniqueness theorem" for first order equations? For nth order equations? Those show that the "fundamental solutons" [b]exist[/b] for each of the "initial value problems" above and the fundamental solutions can then be used to construct the "general solution". Do you know what "linear" and "homogeneous" mean here? Do you understand that the "general solution", as given, only exists for linear homogeneous equations? If not, you are going through the book to quickly and should go back and start again.