The person who taught your epidemiology course is incorrect: As Ilya correctly points out, differential misclassification can certainly occur even in a prospective cohort study. Unfortunately, this exact confusion is very common in epidemiology.

Some reading on how to reason about mismeasurement bias using causal graph is available in Chapter 9 of the Hernan and Robins textbook, which is freely available at http://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/ .The chapter contains all the relevant principles, but doesn't explicitly answer your questions. I also have a set of slides that I use for teaching this material, these slides contain some directly relevant examples and graphs. I can send these to you if you contact me at ahuitfeldt@mail.harvard.edu.

The distinction between "cohort" and "case-control" is not relevant here. The professor is using it as shorthand for retrospective/prospective. The most useful definition of "prospective" and "retrospective" is that in a prospective study, the exposure variable is measured before the outcome variable is instantiated. This is a useful definition because under this definition of prospective, there cannot be a directed path from the outcome to the measurement error on the exposure, which reduces the potential for bias. However, there can still be common causes of the outcome and the measurement error on the exposure, which will results in differential misclassification of the exposure.