I assume that this generator is supposed to generate very narrow impulses? Then the "New Measurement" looks a bit weird. OTOH, the "Old Measurement" actually looks more like it could be the impulse response of the scope's frontend, which is expected to show up on the screen under the assumption that the true stimulus impulse is a very narrow one. The true shape of the stimulus impuse is likely not measurable with this scope anyway.
Risetime is a property of a step response. For an impulse response it is not really meaningful, IMO. You must integrate the impulse response to get the step response, if you want to measure the rise time.