"in" and "out" states in Weinberg's QFT

I think I got the point.

In Heisenberg's picture, the state-vectors do not change according to the Schodinger's equation governing the time evolution of the state. Since different pictures are defined in how operators and state vectors are changing with $\textbf{time evolution equation}$.

But the state vectors $\textbf{do}$ change under symmetry transformations such as Lorentz transformation. And one Lorentz transformation is "time translation", which coincides with time evolution operator of the schdinger equation but the physical meaning is different.

Now let's do a "change of inertial frame of observing the system", what we are doing here is doing a "Lorentz transformation" rather than doing a "time evolution", so the state vectors do change and it changes with the same way of time evolution incidentally.