a simple question I suspect. I have a logistic model with a count predictor (#times some event happened) as a predictor of a binary outcome. Am I correctly interpreting the odds ratio as 'the constant and linear increase in the likelihood of being in the category coded as a '1' on the outcome for each additional event'..?
Pretty much. More precisely, if the coefficient is $a$, then the model says that each occurrence of the event increases the log odds of the outcome by $a$.
Transforming from the logs-odds scale to the odds scale (which is probably easier to imagine), the model dictates that each occurrence multiplies the odds by $e^a$.