Galois extension definition.
We define a Galois extension $L/K$ to be an extension of fields that is
- Normal: if $x\in L$ has minimal polynomial $f(X) \in K[X]$, and $y$ is another root of $f$, then $y\in L$.
- Separable: if $x\in L$ has minimal polynomial $f(X) \in K[X]$, then $f$ has distinct roots in its splitting field.
When $L/K$ is a finite extension, these conditions are equivalent to $L$ being the splitting field of a separable polynomial $f(X) \in K[X]$ - i.e. your condition $1$. This is a fact which is proven in any course in Galois theory. See for example Theorem 3.10 in these lecture notes.
Your condition $2$ is certainly false: for example $\mathbb Q(\sqrt2)/\mathbb Q$ is a Galois extension, but is not the splitting field of $X^5+3X+2$ or of any other (irreducible) polynomial other than $X^2-2$.