You are off a bit on the numbers. First, though, the RIAA suits were not for downloading. The suits were for distribution.
Here is how their enforcement actions generally went.
1. They would initially send a letter asking for around $3 per song that was being shared, threatening to sue if not paid. This typically came to a total in the $2-3k range. There were a few where the initial request was for much more such as when the person was accused of an unusually high volume of intentional distribution. But for the vast majority of people who were running file sharing apps in order to get more music for themselves rather than because they wanted to distribute music it averaged in that $2-3k range.
2. If they could not come to an agreement and actually filed a lawsuit they would pick maybe 10-25 songs out of the list of songs the person was sharing (typically around a thousand) to actually sue over. The range of possible damages in such a suit is $750-30000 per work infringed, with the court (judge and jury) picking the amount [1].
NOTE: it is per "work infringed", not per infringement. The number of infringements will be one of the factors the court will consider when deciding where in that $750-30000 range to go.
3. There would be more settlement offers before the lawsuit actually went to trial. These would almost always be in the $200-300 per song range, which since the lawsuit was only over maybe a dozen or two of the thousand+ songs the person had been sharing usually came out to the same ballpark as the settlement offers before the suit was filed.
Almost everyone settled at that point, because they realized that (1) they had no realistic chance of winning, (2) they had no realistic chance of proving they were were an "innocent infringer", (3) minimal statutory damages then of $750/song x 10-15 songs was more than the settlement offer, and (4) on top of that they would have not only their attorney fees but in copyright suits the loser often has to pay the winner's attorney fees.
4. Less than a dozen cases actually reached trial, and most of those settled during the trial for the same reasons in the above paragraph that most people settled before trial. Those were in the $3-15k range with most being around $5k.
[1] If the defendant can prove they are in "innocent infringer", meaning they didn't know they were infringing and had no reason to know that, then the low end is lowered to $200. If the plaintiff can prove that the infringement was "willful", meaning the defendant knew it was infringement and deliberately did it, the high end is raised to $150k.