Use equation (2), equation (1) is wrong.

Technically the $L^2$ norm (upper case "L") is an integral norm of a function defined as $$ ||f(x)||_2 = \sqrt{ \int_\Omega |f(x)|^2 dx}. $$$$ \left|\left|f(x)\right|\right|_2 = \sqrt{ \int_\Omega |f(x)|^2 dx}. $$ I'm sure you meant $l^2$.

What you typically want to compute in the context of convergence of numerical methods is the finite dimensional analog of the $L^2$ norm which is an area-normalized version of the $l^2$ norm, meaning $$ ||f||_2 = \sqrt{\sum_i |f_i|^2 \Delta x_i} $$$$ \left|\left|f\right|\right| = \sqrt{\sum_i \left|f_i\right|^2 \Delta x_i} $$ Note that $\Delta x$ is inside the square root.

In practice, the area of the domain is just a constant that you don't really care about and the step size ($\Delta x$) is often a constant so you can just use your equation (2). Equation (1) is off by a factor of $1/\sqrt{N}$ so you will underestimate your error for large $N$.