In Hull, we are presented that
$$\frac{\Delta S}{S_{0}}=\mu \Delta t+\sigma\sqrt{\Delta t}\cdot \varepsilon.$$
Following some algebra,
$$ \begin{align*} \frac{\Delta S}{S_{0}} &=\mu \Delta t+\sigma\sqrt{\Delta t} \cdot \varepsilon \\ \frac{S-S_{0}}{S_{0}} &= \mu \Delta t+\sigma\sqrt{\Delta t} \cdot \varepsilon \\S &= \left(S_{0} + \mu S_{0} \Delta t\right) + \sigma S_{0} \sqrt{\Delta t} \cdot \varepsilon \end{align*} $$
Therefore the distribution of future stock price is given by
$$S \sim \phi\left(S_{0} + \mu S_{0} \Delta t,\left(\sigma S_{0} \sqrt{\Delta t}\right)^{2}\right).$$
That is, the future stock price follows a normal distribution.
We are then introduced to Itô's Lemma. By letting $G = \ln(S_{0})$, we derive that
$$dG = \left(\mu - \frac{1}{2}\sigma^{2}\right)dt+\sigma dz.$$
Since $G = \ln{S_{0}}$, in a discrete sense, it can be said that
$$dG = \ln{S_{T}} - \ln{S_{0}}.$$
Therefore,
$$\ln{S_{T}} - \ln{S_{0}} = \left(\mu - \frac{1}{2}\sigma^{2}\right)dt+\sigma dz \\ \implies \ln{S_{T}} = \ln{S_{0}} + \left(\mu - \frac{1}{2}\sigma^{2}\right)dt+\sigma dz. $$
It then follows that since $\ln{S_{T}}$ follows a normal distribution, the future stock price must follow a lognormal distribution
I am now confused, which process do I use to answer questions about the probabilistic nature of future stock prices?
I have one other question. Why does
$$\mathcal{P}(\ln{S_{T}} > \ln{X}) = \mathcal{P}(S_{T} > X)?$$
The context for my last question can be found here.