I think that it shouldn't be included. To me, a turnover is an indication that a player made some sort of mistake from an offensive perspective or a great play from a defensive perspective.
Turnover on downs is an indication that the coach's play calling was the more significant problem. 4 plays called didn't get you ten yards rather than one play's mistake/great play allowed the other team to attain possession.
The halftime stats on the jumbotron listed us with 2 turnovers, when we only had one. I too was wondering if they counted the failed fourth down attempt as a turnover.