The issue of "mid-P" values and confidence intervals is complex and frequently misunderstood. It arises only with discrete distributions (in StatsDirect with proportions and binomial distributions).
In short, "exact P" methods attempt to align alpha with the maximum type I error rate and 1-alpha with the minimum coverage, whereas "mid P" methods try to align alpha with the mean type I error rate and 1-alpha with the mean coverage. Remember that alpha = significance = probability of type I error and type I error = the false rejection of the null hypothesis (see P).
Statisticians do not agree universally upon when to use "mid P" methods. Armitage and Berry recommend using "mid P" methods when comparing several trials (e.g. 95% confidence intervals for the difference between two independent proportions from ten similar trials) and to use "exact P" methods when considering single trials in isolation.