What's the oldest version that I should try that has a different counting system?
One part of the counting mechanism has been modified for version 8.8.5 or 8.8.6. But that was only a minor improvement and no modification of the method. So it is save say that the counting mechanism has not changed since 8.8.
Independent of that, it is recommended to downgrade from 8.8.10.x to a version prior to 8.8.10. Because the organisation of the postmeta information of podPress has changed with 8.8.10 (see the changelog). You would need to adjust these datasets manually if you would downgrade. Otherwise no podPress attachements would appear in posts an feeds.
The most significant change since 8.8 which influences the download numbers is probably the the way modern browsers like Chrome and Safari handle the pre-loading of media files in cases when the media files are embedded with HTML5 tags (
<video>) and the integration of HTML5 support into podPress.
If a listener goes to a blog with podPress 8.8.10 - 18.104.22.168 while he/she uses e.g. Chrome and the option "Use HTML5 tags:" is checked then a click on the play button would be counted multiple times because Chrome requests the file multiple times if the files was not played before in the same session. (Safari request the file in the same situation only one time.)
If a podcaster would also activate "show HTML5 players always on page load" then the numbers would be even higher. Because the desktop versions of Safari and Chrome would pre-download (buffer) all media files which are embedded with HTML5 tags on page load. In the statistics it would look like someone clicked multiple times on all the players of the recent podcast episodes. I have read tweets of podcasters which have mentioned this effect basically.
But I have understood the problems after you have reported the problem you have encountered. (Thanks again for reporting the problems here!)
in conclusion: with 22.214.171.124 the numbers would be higher depending on your settings and the number of users which are using Chrome, Safari or IE9.
But I have no explanation for the obvious sharp bend in your statistic. Also I could not reproduce this problem in my own blog. That is why I'm not sure whether it is really a problem of podPress or not. As I have written above the counting mechanism has not change in a long time and the one thing which may have influenced the numbers since 8.8.10 is fixed now.
Maybe control the raw data. Are there a lot of downloads from in a row from the same IPs? Have you blocked the search engines some how in that time when you have installed 8.8.10. the first time? or did you modified other settings of podPress or your blog?