I'm struggling with what the exact meaning of "stock prices are lognormal" (and its use to show normality of returns). My assumption was that given ${S_t}$ are stock prices and returns are defined as $r_t = \frac{S_t-S_{t-1}}{S_{t-1}}$, then we assume $S_t$ is lognormal, and then:
$$\log(1+r_t)=\log\left(\frac{S_{t-1}+S_t-S_{t-1}}{S_{t-1}}\right)=\log\left(\frac{S_t}{S_{t-1}}\right)=\log(S_t)-\log(S_{t-1}) \tag{1}$$
As this would be the sum of two normal variables, the result is normal, and that allows us to show $\log(1+r_t)$ is normal.
However, I was reading the following link:
In it, the author states (I've replaced his notation with mine for ease of comparison):
If we assume that prices are distributed log normally (which, in practice, may or may not be true for any given price series), then $\log(1+r_i)$ is conveniently normally distributed, because:
$$1+r_i=\frac{S_{t}}{S_{t-1}}=\exp\left(\log\left(\frac{S_t}{S_{t-1}}\right)\right)\tag{2}$$.
From the definitions of lognormal, in order for the inner term of the right-hand side (i.e. $\log\left(\frac{S_t}{S_{t-1}}\right)$) to be normal, we would need $1-r_i$ to be lognormal. But that seems different to me than "prices are lognormal". The following cross validation answer makes a bit more sense of this, namely part ii), where the answerer mentions conditional lognormality, or that the assumption of log normality in prices usually refers to $\frac{S_t}{S_{t-1}}$, and that would satisfy equation 2.
So to summarize, what is the correct way to define the lognormality assumption in prices? My apologies if I'm simply overthinking things. Thank you!