What should define a SARS-CoV-2 "breakthrough" infection?

Recently there have been several reports of SARS-CoV2 "breakthrough" infections that have occurred in recipients of the FDA approved SAR-CoV-2 vaccines. The use of the term "breakthrough" infections implies that the virus broke through a protective barrier provided by the vaccine. However, is this what happened in these cases? In most cases, the answer is no, and this answer lies in the fundamental understanding of the mucosal immune system. Here we suggest a more precise definition of what a true breakthrough case is.