The concept of greater and lesser jihad is taught fairly commonly in Western schools nowadays. I learned about it my AP world history class, and this comment near the top of the ELI5 "What is jihad" thread discusses. However, this source as well as comments by my history teacher leads me believe that this idea is not widely accepted, and is likely introduced to make jihad seem more amicable toward modern moral values.

What is the history of this interpretation and other similar interpretations of jihad? Is this idea widely accepted? How did and does the idea of jihad reconcile with the historic semi-tolerance and tolerance of other religions that has existed in most Islamic areas of the world?