How could the whiskers be minimum and maximum if my first example is showing an outlier that is less than the lower whisker? From my knowledge the standard method for BW plots is to use the tukey method (1.5*IQR) to determine fences and anything outside that are outliers.
They are whichever is closer to your data set (i.e. if your max/min are smaller than 1.5 x IQR then use max/min, if they are greater, they are considered outliers).