Impulse and unit step response
In response to MBaz's comment, if we define the impulse response (edit) for Linear Time Invariant systems as 


  "the function $h(t)$ such that the mapping $x(t) \mapsto y(t)$ is computed as $y(t) = x(t) * h(t)$"


then notice that if $x(t) = \delta(t)$ you get $y(t) = \delta(t) * h(t) = h(t)$.

Thus louism's statement a) and MBaz's comment agree.