"the amount of time that combat was going on, in which you actually did things":
If combat lasts a minute, and you hit buttons for the whole minute, your active time is one minute.
If combat lasts for a minute, and you hit buttons for thirty seconds, then sit on your hands, your active time is thirty seconds.
In the first case "effective" DPS and "active" DPS are identical. In the second case "effective" DPS is exactly one half of "active" DPS: recount gives you "total damage you did divided by the amount of time not sitting on your hands doing nothing".
Yeah, what you describe is exactly "effective" DPS: how much damage you did over the whole course of the fight, rather than just how much while you were "active" (eg: hitting buttons).
Ultimately, "effective" DPS is a measure of your contribution to the raid, and "active" DPS is a measure of how hard you hit while doing stuff. The former is, in PVE (and, IMO, PVP but much more debatable there) the most useful measure of how helpful your contribution was.
After all, if you spend ten seconds moving out of mechanics and I spend thirty, your contribution to the overall effectiveness of the group is legitimately higher -- even if my "active" DPS while not running around is higher than yours, right?
Likewise, if I spend way more time CCd than you because I don't have a trinket, or waste it, then my overall contribution is lower than yours even if my uptime is higher when I am not locked down.
PvP is more complex than PvE, of course: if the other team focus me exclusively and I stay alive just to spite them, my eDPS will be low, but my contribution high -- this single number can't capture the whole picture. Ditto, if you are on "run around for some mechanic" duty in PVE -- engineers on Garrosh, or orbs on twins -- then maybe this number is low, but without you the raid would fall apart.
Which is why people often say that a *good* raid leader or pvp team don't just use DPS numbers as their measure of skill, they take the whole picture into account.