From my studying difference equations and transfer functions, I understand that when a complex exponential input $x[n]=z_1^n$ is applied to an LTI system with transfer function $H(z)$, determining the output involves evaluating the transfer function at $z=z_1$, i.e. $H(z_1)$ which is a complex quantity that simply scales the input signal to produce the output. The textbook I have uses a sinusoid input to show the effect (the transfer function causes a magnitude and phase change to the input).
The answers to this question I asked a while ago made the point of recognizing the difference between $a^n$ and $a^n \cdot u[n]$. All of that was in the context of two sided signals and the bilateral z-transform.
In something I found online, in the context of causal signals and the unilateral z-transform, there was a question posed: "if $x[n]=(1/2)^n \cdot u[n] \to y[n]=\delta [n-2],$ what is the output for $x[n]=\cos(\frac{\pi}{3}n)?$", where the author determined the output by evaluating $H(z)$ determined from the first relationship at $z=e^{jn \pi /3}$. Since this is a 'causal' context, isn't that cosine really $x[n]=\cos(n\frac{\pi}{3}) \cdot u[n]$ and hence requires taking the z-transform of the input?
When can you just evaluate $H(z)$ at the value of interest, and when do you have to take the (unilateral) z-transform of the input?