Reporting odds ratios from a (non-logistic) binomial regression I ran a aggregated binomial regression, with a non-binary bounded count variable, number of counseling sessions (out of a maximum possible six) attended during a clinical trial as the outcome. So participant 1 may have attended 0/6 sessions, participant 2, 4/6, participant 3 1/6 and so on.  This was not a poisson regression which is for count outcomes where there is no known upper limit for the count.
The exponentiated log-odds coefficient (i.e. the odds ratio) for the predictor 'treatment group' was 1.43.
Now if this was a binary/bernoulli outcome I would report this with something like 'the odds of people in the experimental group attending a counseling session were 43% higher than for people in the placebo group'.
But how do I do this for a non-binary bounded count outcome (i.e. a count outcome where the maximum possible count is known)? It's still an odds ratio but I am just not sure how to word it. All the online guides for reporting are for logistic regression with bernoulli-type outcomes.

From the question and comments, this is simply a binomial logistic regression, as @jwimberley suggests in a comment.
You are modeling the probability of attending a counseling session, with each individual given 6 chances to attend and with no attempt to model differences in attendance over sessions/time. That's modeled as a binomial distribution, the distribution of the number of successes over a defined number of trials with some constant probability of success. There's no need to restrict the data setup into the form of individual 0/1 Bernoulli trials
The details of how you did the analysis aren't clear, but in R you could have properly used either of the following ways to present such data to glm() for a logistic regression (see the R manual page for family() in the stats package):

As a numerical vector with values between 0 and 1, interpreted as the proportion of successful cases (with the total number of cases given by the weights)
As a two-column integer matrix: the first column gives the number of successes and the second the number of failures.

That allows the model to take the total number of observations into account. You can report log-odds or odds ratios however you wish. There's no need to qualify this as a "weighted" regression. It's just that you presented the data to the software in binomial rather than Bernoulli form.
