Logistic regression models the log odds of an event as some set of predictors. That is, $\log(p/(1-p))$ where p is the probability of some outcome. Thus, the interpretation of the raw logistic regression coefficients for some variable ($x$) has to be on the log odds scale. That is, if the coefficient for $x = 5$ then we know that a 1 unit change in x correspondents to 5 unit change on the log odds scale that an outcome will occur.
However, I often see people interpret exponentiated logistic regression coefficients as odds ratios. However, clearly $\exp(\log(p/(1-p))) = p/(1-p)$, which is an odds. As far as I understand it, an odds ratio is the odds of one event occurring (e.g., $p/(1-p)$ for event A) over the odds of another event occurring (e.g., $p/(1-p)$ for event B).
What am I missing here? Is seems like this common interpretation of exponentiated logistic regression coefficients is incorrect.