Scaling the input vs scaling the impulse response for an LTI system
Aside from

Input that is only a constant value or
a System that does nothing to the signal, i.e. $H = \text{const.}$,

these things are not the same.
Sketch of a proof:

Scaling input shifts energy to frequency components that hadn't energy before,
Scaling the system means changes the Transfer Function, but keeps the system LTI, which means it can't add energy to any components that had none before.