That even so, afterward I counted out loud the wrong number, and the day has already passed. For example, in the evening I counted 13, and then during the day I thought I had gotten mixed up and counted 14, and then the evening of 14 arrived. Should I count with a blessing, even though I was under the mistaken impression, or since it was actually mistaken does my counting continue, since I did count, even though that wasn’t what I had in mind at the beginning of the fourteenth day?

It doesn’t seem to me that a mistake during the day ruins a correct count at night. It should be remembered that, in any case, the whole issue of not counting with a blessing if you miss a day is itself a dispute among the medieval authorities (Rishonim).
However, one could discuss a case where in the evening too you thought it was 14, but got mixed up and mistakenly counted 13 instead (a double mistake). Here, in my opinion, you really did miss the day.