Definition $1$:
A polynomial over $F$ is called separable if it has no multiple roots (i.e., all its roots are distinct).
Definition $2$:
An irreducible polynomial $f(x)\in F[x]$ is separable over $F$ if $f$ has no repeated roots in any splitting field.
Questions:
$(a)$ What is the meaning of $S=\{f_i(x):f_i(x)\in F[x]\enspace \forall 1\leq i\leq n\}$ to be separable over $F$ according to the definition #$2$ ?$($ $S$ is not a multi-set$)$.
$(b)$ Is definition #$1$ weaker than #$2$ ? If not, Show the equivalence.
Few Comments:
For part $(b)$, I guess, they are not equivalent. Reason: Definition #$2$ requires the extension to be splitting field whereas #$1$ can have any extension in which the polynomial splits.
Moreover, I can make a sense of $S$ to be separable over $F$ according to the definition #$1$ but it doesn't make sense in definition #$2$ because the splitting field for each polynomial in $S$ is different.
The first definition is nonsense. The polynomial $X^p-t$ over $F(t)$, $t$ an indeterminate over the $p$-element field $F$, has no roots at all in $F(t)$, and according to definition 1 it would be separable.
Actually a polynomial is called separable if its irreducible factors are according to definition 2. For instance, $(X-1)^2$ is separable.
Note that having multiple roots in any splitting field can be checked in the field where the polynomial is defined in, so the choice of the splitting field is irrelevant.
Theorem. Let $F$ be a field and $f\in F[X]$ a polynomial. There exists an extension field $K$ of $F$ where $f$ has multiple roots if and only if $f$ has a non trivial common factor with its derivative polynomial $f’$.