The probability of morale appearing depends very much on the number of restarts!
For example, with a morale of +3, the probability of it appearing in two battles is already 23.4%, in three - 33%, which is significantly different from the 12.5% in your manual. And if you take into account targeted "catching," then it's even 100%!

Well, of course, we are talking about different things; you seem to be familiar with mathematics, so you should understand the difference between the probability of each specific event occurring (12.5%) and a series of events in a row...

Moreover, we are not considering the battle as a whole, but each turn during the battle. Perhaps this is what confused you in the manual. With a probability of 12.5%, on average, morale will appear once every 8 rounds of the battle for each unit. I highlighted the key word "on average," as morale may appear three times in a row, and then not appear at all for a long time... Your statement is either incomplete (and I simply didn't catch what was meant), or it distorts the essence of the issue...