That probably depends on what types of jobs you are applying for.  If you are applying only for temporary summer jobs, then you are committed to returning to the teaching job in the fall (just like any other teacher who has the summer off) and you are not unemployed.  If you are looking for and would accept a permanent, fulltime position with another employer and if you are applying for those sorts of jobs, then I think you have a valid argument that the partime job in the fall is just a fallback if you don't find something better.  If so, I would apply for unemployment.