Well, working off of a word that is probably more well known -- "semi-annually" -- which means twice a year, it would seem to indicate that semi-monthly would mean twice a month, and bi-monthly would then mean every other month. That's the way I learned it, too -- Miriam notwithstanding.
I also work for attorneys, and they must fulfill "x" number of Continuing Legal Education (CLE) hours per two-year period (known as a biennium). So since a biennium (or bi-annually) means every other year, ditto for bi-monthly -- every other month. As one other example, a bi-monthly newsletter is sent every other month.
And yes, we do have a screwy language -- ask any non-English speaking person who has had to learn our language.