I'm reading Stochastic Differential Equations (sixth edition) by Bernt Oksendal and I'm trying to understand the author's definition of the Ito integral. For a given probability space $(\Omega, \mathcal{F},P)$, the author defines the Ito integral as follows:
Definition 3.1.6 (The Ito integral) Let $f \in \mathcal{V}(S,T)$. Then the Ito integral of $f$ (from $S$ to $T$) is defined by $$ \hspace{5mm} \int_{S}^{T} f(t,\omega) \, dB_t(\omega) = \lim_{n \to \infty} \int_{S}^{T} \phi_n(t,\omega) \,dB_t(\omega) \hspace{1cm} (\text{Limit in } L^2(P)) \hspace{5mm} (3.1.12) $$
where $\{\phi_n\}$ is a sequence of elementary functions such that $$ \hspace{2cm} \mathbb{E} \left[ \int_{S}^{T} \left( f(t,\omega) - \phi_n(t,\omega) \right)^2 \,dt \right] \to 0 \quad \text{ as } n \to \infty. \hspace{2cm} (3.1.13) $$
What exactly does equation (3.1.12) mean? I'm a bit unsure about the meaning of the "limit in $L^2(P)$" part. My best guess is that it means the following: $\omega \mapsto \int_{S}^{T} f(t,\omega) \,dB_t(\omega)$ is the (unique) random variable on $(\Omega,\mathcal{F},P)$ satisfying $$ \lim_{n \to \infty} \mathbb{E} \left[ \left( \int_{S}^{T} f(t,\cdot) \,dB_t(\cdot) - \int_{S}^{T} \phi_n(t,\cdot) \, dB_t(\cdot) \right)^2 \right] = 0. $$
Is this what the author means?