The form of the rule that I was told in school is: I before E, except after C, when the sound is "ee". This (1) is pretty simple, (2) has rather few exceptions, and (3) does correctly give the spelling of lots of words that some people would otherwise find tricky.

Note that "except after C" means "ignore the rule in this case" rather than "reverse the rule in this case". That is, it doesn't mean "E before I, after C".

Of the 9 words on your list: 5 (science, frequencies, vacancies, fallacies, society) are "after C" and therefore not covered by the rule; 2 (eider, feisty) are never pronounced with an "ee" sound and therefore not covered by the rule; 1 (either) is sometimes pronounced that way and sometimes not and therefore a kinda-half-exception; 1 (seize) is a definite genuine exception.

There are a few other exceptions, but the rule in this form correctly fixes the spelling of a whole lot of common words (a few examples: piece, frieze, grief, believe, retrieve) that would be easy to get wrong without it.