Principle of stationary action vs Euler-Lagrange Equation

Granted appropriate boundary conditions, the stationary action principle and the Euler-Lagrange (EL) equations are both precisely the condition that the functional/variational derivative $$\frac{\delta S}{\delta x^j (t)} \tag{1} $$ vanishes, so they better agree!

First, I think there is something wrong with your partial derivative of the Lagrangian with respect to $x$.

Second, the Euler-Lagrange equations are nothing more than the process that you performed in Method 1, done without committing to a specific form for $L$ but leaving it generic. In your first step you took partial derivatives of $L$ with respect to its position and velocity terms, in your second step you took the velocity derivative and involved it in an integration by parts, where you took a total time derivative and then added a minus sign. If your Lagrangian also involved $\ddot x$ you would then have a $+\frac{\mathrm d^2~}{\mathrm dt^2}\frac{\partial L}{\partial\ddot x}$ term from two integrations by parts, for example.