Using "Stop Time" option prevents play count from being updated
It's somewhere between a bug and not-a-bug. Play count increments when the last sample plays (or something roughly equivalent). But the duration metadata options don't actually change the file, so the very end still exists, but is always skipped. So, the play count never increments.

The only workaround I know involves changing the audio file itself. 

If you're using mp3s (or AAC or another lossy codec), doing this in iTunes involves re-encoding (transcoding) an already lossy file, a minor pet-peeve of mine and something I do not recommend. I suspect there are paid audio editing tools which can cut an mp3 without further losses, but I don't know what they might be.

If you're using apple lossless (or wav), or you really don't care about transcoding, you can do it simply enough in iTunes:


I'm assuming you've already set the start and stop time metadata appropriately. 
In iTunes, go to Preferences -> General -> Import settings... and set everything to match the encoder settings of the original file (you can see this info by choosing "Get info" on the song in question).
Select the song, right click it and choose "Create [codec] Version", where [codec] is whatever you selected (for instance, if you chose ALAC it will say "Create Apple Lossless Version").
Done. iTunes has truncated the audio file to the limits set by your start/stop time metadata.