Deionized water usually contains covalently bonded organic chemicals as minor components and traces. Distilled water usually contains dissolved ORP and acid gasses like Cl2, CO2, SO2, or NOx that result from economies of condensers. In either case when water quality is analyzed in laboratories the test equipment is usually rinsed with a portion of the sample shortly before an analysis is done. It prevents false results in highly regulated water processes.. 

Distilling deionized water gives a more pure condensate than DI water or distilled tap water.  Example is USP water for injection where a number of purification steps are taken to get the required purity.

You will probably get a small carryover from the trace organics left from the DI media. I would suggest a carbon filter of the DI water prior to distillation.