Some may wonder, how can these be synonymous, if "off" and "on" have the opposite meaning?
The key word in the Wiktionary link is nonstandard, which can be loosely translated as: yes, you'll hear it, but yes, there's a better way to say it.
Also, some prepositions in such phrases are incidental, and can fluctuate. You might say:
This movie was based on a true story.
This movie was based upon a true story.
This movie was based off of a true story.
All would have the same meaning, although the third has an amateurish ring to it.