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.