Overlapping samples

I am analyzing a cohort of patients where we use two different definitions for a specific disease. The patients are coded either as having the disease or not according to each definition. There is a relevant overlap of patients coded as having the disease according to both definitions, and a smaller part of patients each only coded as having the disease according to only one of the two definitions. Now, I would like to test if there is a significant difference in mortality between the two groups according to the two different definitions.
How would you do that? Would you only compare the difference of the two non-overlapping groups (e.g. in a cross-table with a fisher T-Test), excluding all overlapping patients (but I think then I am loosing statistical power). Or is there another possibility of comparing both groups including the overlapping (double-positive coded) patients?
Thanks for some help.