Given a pathogen and a corresponding vaccine, and given an immune response to either one that results in antibody creation, would it be expected that a clinical test or assay for said antibodies would find it difficult (or SHOULD find it difficult) to distinguish between antibodies generated from exposure to the live pathogen vs a well-designed vaccine?
Or asked another way, would it be the goal of a well-designed vaccine to elicit an immune response that would generate antibodies that are indistinguishable from those generated by exposure to the actual pathogen? Such that an antibody test would test positive in either case?
Or perhaps more to the point - are there examples of vaccines (prior to 2019) that lead to the creation of antibodies that are not detected by clinical tests for naturally-induced antibodies (where such tests exist) ?