The Lagrange density is ($c=1$)
\begin{equation}
\mathscr L\left(\phi\left(\boldsymbol x,t\right),\boldsymbol\nabla\phi(\boldsymbol x,t),\dot{\phi}(\boldsymbol x,t)\right)=-\frac12\left(\dot{\phi}^2-\Vert\boldsymbol\nabla\phi\Vert^2\right)-V\left(\phi\right)
\tag{01}\label{01}
\end{equation}
while the Lagrangian is
\begin{equation}
L(t)=\int\mathrm d^3\boldsymbol x\,\mathscr L\left(\phi\left(\boldsymbol x,t\right),\boldsymbol\nabla\phi(\boldsymbol x,t),\dot{\phi}(\boldsymbol x,t)\right)
\tag{02}\label{02}
\end{equation}
The Euler-Lagrange equation in terms of the Lagrangian and its functional derivatives is
\begin{equation}
\dfrac{\partial}{\partial t}\left[\dfrac{\delta L(t)}{\delta\,\dot{\!\phi}(\boldsymbol x,t)}\right]-\dfrac{\delta L(t)}{\delta\phi(\boldsymbol x,t)} =0
\tag{03}\label{03}
\end{equation}
Between the functional derivatives of the Lagrangian $L(t)$ and the partial derivatives of the Lagrange density $\mathscr L$ we have the following relations
\begin{align}
\dfrac{\delta L(t)}{\delta\,\dot{\!\phi}(\boldsymbol x,t)} & =\dfrac{\partial \mathscr L}{\partial\,\dot{\!\phi}(\boldsymbol x,t)}
\tag{04a}\\
\dfrac{\delta L(t)}{\delta\phi(\boldsymbol x,t)} & =\dfrac{\partial \mathscr L}{\partial\phi(\boldsymbol x,t)}-\boldsymbol{\nabla\cdot}\left[\dfrac{\partial \mathscr L}{\partial\left(\boldsymbol\nabla\phi(\boldsymbol x,t)\right)}\right]
\tag{04b}
\end{align}
so the Euler-Lagrange equation in terms of the Lagrange density and its partial derivatives is
\begin{equation}
\dfrac{\partial}{\partial t}\left[\dfrac{\partial \mathscr L}{\partial\,\dot{\!\phi}(\boldsymbol x,t)}\right]+\boldsymbol{\nabla\cdot}\left[\dfrac{\partial \mathscr L}{\partial\left(\boldsymbol\nabla\phi(\boldsymbol x,t)\right)}\right]-\dfrac{\partial \mathscr L}{\partial\phi(\boldsymbol x,t)} =0
\tag{05}\label{05}
\end{equation}
