Bounded weak and weak-$star$ topologies and metrics

Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set
$$
d(x,y) := sum_{n=1}^infty 2^{-n} |(x-y, h_n)|,
$$
where $(x-y, h_n)=(x-y, h_n)_{X,X^*}$ is the dual pairing on $X$.
Then $d$ is symmetric, satisfies the triangle inequality and $d(x,y) = 0 implies x = y$, hence $d$ is a metric.

On closed bounded (by the norm) balls of $X$, the metric $d$ induces the weak topology.

Can the topology induced by $d$ be described in a functional way in relation to the weak topology or the bounded weak-topology? Would any answer change if $d$ is defined using the ratio $frac{|(x-y,h_n)|}{1 + |(x-y,h_n)|}$ ?

This seems to be something that should be in Megginson’s book on Banach spaces, I can only see something related in exercise 2.86-88 of section 2.7.