The definitions are not equivalent. As far as I know, the usual definition is the second one:
Definition: The directional derivative of $f$ at $p$ in the direction of the unit vector $\vec{s}$ is the derivative at $0$ of the function $\varphi(t)=f(p+t\vec{s})$
The first "definition" in the question is actually a property that is not equivalent to the definition. It is true for example if $f$ is differentiable at $p$:
Theorem: If $f$ is differentiable at $p$, then $f$ has directional derivatives at $p$ in every direction $\vec{s}$ and $f_{\vec{s}}(p)=\nabla f(p)\cdot \vec{s}$.
The proof is simply a consequence of the Chain Rule for differentiable functions. If $f$ has all partial derivatives (hence has a gradient) but is not differentiable, things get messy. The RHS is defined but it is possible that the LHS does not exist for some direction $\vec{s}$. Worse, it is possible that $f_{\vec{s}}(p)$ exists but is not equal to $\nabla f(p)\cdot \vec{s}$!
Some interesting examples (at $p=(0,0)$):
- $f(x,y)=0$ if $x=0$ or $y=0$, and $f(x,y)=1$ otherwise.
- $f(x,y)=\frac{y^2}{x}$ if $x\neq 0$ and $f(0,y)=0$.
Those examples are classic Calculus/analysis textbooks material. In some contexts (differential geometry?), we don't care about these kinds of pathological examples and consider only differentiable functions, so the formula is always true.