Computing the density of the smallest and second smallest values
Your error is that you cannot say that the events
$E_1:=(X_{(1)} \leq t_1, X_{(2)} \leq t_2)$ and $E_2:=(X_{(1)} \leq \min\{t_1, t_2\})$ are the same.
You can just say that $E_1 \subset E_2$ (meaning $E_1 \implies E_2$), but $E_2$ cannot imply $E_1$ ; therefore you haven't the corresponding equality of probabilities.