Power is always energy per time, J/s, aka W. If it's moving, you get less energy into one spot because of the shorter time, but the power (energy per time) will still be the same. What you're looking for is energy.
As far as safety, it's both. It's how much energy you put into a spot on your eye. The energy input is what damages cells. But with higher powered lasers, the time it's on a spot on your eye may be very short, but the energy imparted could still very easily be enough to hurt your eye.
As far as average power vs. peak instantaneous power: The peak instantaneous power will the the power of the laser. The average power will be the fraction of the time the laser spends on that spot multiplied by the power of the laser. Say: 100mW laser spends 1% of the time on a particular spot, average power is 1mw. But, peak instantaneous power is still 100mW, and that is still very damaging. For what you're asking, I think you should be more worried about the peak instantaneous than the average over time, because that peak can do damage where that average over time may not.
For example, let's say I use a 250mW laser for one minute a day. Over the course of a day, that's .06% of the day. That means the average laser power over the day would be .17mW. That average power, while still low enough to look safe, doesn't imply that using the laser for only one minute a day is still safe because the average is lower, especially since a 250mW can do damage in a shorter time than the blink of an eye. The peak instantaneous power of the laser would do damage, even though the average power is very low.
At least I think it answers the question you are asking. Every situation is different though. You're asking about short flashes of a higher-powered laser. Another situation may be about very long exposures to dim laser light, like 5mW constantly all day in a lab, and that would call for a different evaluation of what to check for. But this stuff has all been looked at, there are standards/guidelines to go by for safety glasses for what is good enough for short flashes of light, and what is good enough for all-day exposure to laser light. FrothyChimp could contribute a lot more about such things.
