Is this HW question even valid? (linear systems)
there is an SE specifically for signal processing.  sorta surprized to see tags for this at the Physics SE.

what you are suspecting is precisely true.  LTI is a property of the system, not of the input.  both 1. and 2. are obviously linear and time-invariant.

one slight misconception.  in DSP we usually use square brackets, as in "$x[n]$", to depict discrete-time signals (as an alternative to "$x_n$").  a discrete-time signal is just a sequence, but we want to leave the subscript for the same purpose we might with continuous-time signals.  i.e. given a vector of continuous-time signals: $\{x_1(t), x_2(t), ... x_N(t) \}$, if we were to sample those signals, the corresponding discrete-time vector would be $\{x_1[n], x_2[n], ... x_N[n] \}$.  the argument (inside of square brackets) to a discrete-time signal is always an integer.

that said, "$\delta[n]$" is the Kronecker delta, not the Dirac delta.  (the latter the math guys like to tell us is not even a "function".)

$$ \delta[n] \ \triangleq
\begin{cases}
1, &amp; \text{if }n = 0 \\
0, &amp; \text{if }n \ne 0
\end{cases} $$