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.
If future generations are to remember us with gratitude rather than contempt, we must leave them more than the miracle of technology. We must leave them a glimpse of the world as it was in the beginning, not just after we got through with it.
- Lyndon Johnson, on signing the Wilderness Act into law (1964)