As @Brent Kerby said
[..] the increments of a GBM are neither stationary nor independent.
This is the reason why GBM is not a Lévy process. What I instead proved is the non-stationarity of the process itself, which is not taken into account by the definition of Lévy process.
Note
The use made of Lévy processes in modern quantitative finance is the following: instead of using the GBM as the stochastic process followed by the stock prices (as in the Black-Scholes model), different stochastic processes are taken into account to describe the dynamics of the stock prices. Many of this others stochastic processes are Lévy processes.
The main example is the Variance Gamma process, which can be written as a time-changed Brownian Motion ${W_T}_s$ subjected to an independent increasing jump process, a so-called Gamma Lévy process with $T_s \sim Gamma(\alpha s, \beta)$. The process ${W_T}_s$ is then also a Lévy process itself.
The advantage of using Variance Gamma process instead of GBM to model stock prices is that the former takes into account GBM problems such as the Gaussian density decreasing too quickly, absence of variation of the volatility $\sigma$ over time, absence of jumps.
Best Answer
Perhaps it's the use of $\mu$ in both formulas that is confusing you? The first reference gives the definition of geometric Brownian motion as $$\frac{dS_t}{S_t} = \mu dt + \sigma dW_t$$ and the second as $$\frac{dS_t}{S_t} = r dt + \sigma dW_t$$ so $\mu$ in the first reference is $r$ in the second. The second reference then uses $\mu$ as a notation for $r-\frac{1}{2}\sigma^2$. So in other words, the symbol $\mu$ in the second reference does not have the same meaning as the symbol $\mu$ in the first reference. But both are mathematically the same.