Webster came through with a wonderfully thorough definition, and an exact timing of when this occurred (over a century ago!!!). I guess my old dictionaries were remiss not to include both definitions. Here's the link to see their explanation.

http://www.m-w.com/dictionary/bi-

Most dictionaries have always defined “bi” as both two and twice; perhaps this will add to the confusion and incredulity. The below conclusion unfortunately does not account for "biannually", which always means twice a year (verses biennially, meaning every other year). I have conducted some superficial research, and it appears as though this shift occurred between 1970 and 1980, and was brought about by the publication industry (I guess it just took too much printing space to use every other week and every other month). I’m afraid this is something with which I will have to live, considering meanings/definitions of words can and will slowly change through time……Mara is correct in this statement.