What app you use to download an app shouldn't matter. However what feed you use might. Most podcast apps do things the right way--they just direct the user to the feed the podcast provides, meaning the podcast provide can track its own downloads. But Stitcher, which is pretty popular, does not; it re-hosts things itself, which is stupid.
This is the actual feed:
http://thebestshow.libsyn.com/rssAny good iOS (Pocket Casts, Instacast, Downcast, Overcast) or Android (Overcast) app's downloads should count things the right way.