Strong and weak-* convergence for bounded linear maps

The textbook I am using says that a sequence $(T_n)$ in $\mathcal{B}(X,Y)$ for $X$, $Y$ normed linear spaces converges strongly to $T$ if $\lim_{n\rightarrow\infty}T_nx=Tx$ for every $x\in X$.

The (topological) dual space to $X$ a Banach space is defined as $X^\ast=\mathcal{B}(X,\mathbb{R})$, and $\varphi\in X^\ast$ is the weak-$\ast$ limit of $(\varphi_n)$ if $\varphi_n(x)\rightarrow\varphi(x)$ as $n\rightarrow\infty$ for every $x\in X$.

This seems like the same definition to me. Is the difference that we are considering $X^\ast$ to be a space in its own right, and then by strong convergence in $X^\ast$ we actually mean norm convergence, where $\|\varphi\|=\sup_{x\neq0}\frac{|\varphi(x)|}{\|x\|}$?

And of courese you are right that weak-$^*$ convergence is the same as strong operator convergence in $X^*$. These are just different names for the same thing. Of course when one deals with functionals he uses notion of weak-$^*$ convergence. The term strong operator convergence is used for "real" operators.