Using a Poisson rate, your prevalence is 5.93 per thousand, with a CI of 4.67 to 7.44. The standard error is 0.685.
I have no idea how you did the calculations above, but a Poisson rate does not have a standard deviation, and I have no idea what you mean by 'weighted mean' either.
Calculations done in Stata 15
. cii means 12.642 75, poisson
-- Poisson Exact --
Variable | Exposure Mean Std. Err. [95% Conf. Interval]
Weighted mean is different from arithmetic mean. The weighted mean is used when variables have different impacts or weight to an intervention while arithmetic mean is preferred when they have equal impacts or weight. Example , when you are analysing class scores of a student to overall scores of many subjects, it is better to use weighted means so as to understand his strength hence weighted means of the total scores. In case of prevalence where the factors under consideration are equal in contributions, i will prefer arithmetic normal mean.