Interpretation of 'recall' as a conditional probability P( X=>+ | X=+ )
Your interpretation is correct.
$ F_n $ represents the number of false negatives whereas $ F_p $ represents the number of false positives.
Recall
answers the question:
when presented a positive example, how often does the classifier get it right?
aliases: sensitivity, true positive rate (TPR)
Precision
answers the question: out of all the examples the classifier thought were positive, how often were the examples actually positive?
aliases: positive predictive value (PPV)