It seems to me that this confusion is perpetuated by “authorities” who have accepted the generally misunderstood use of the words. This definition below actually has some merit and should be taught by all:

http://www.elearnenglishlanguage.com/difficulties/bisemi.html

The English prefixes bi- and semi- are often mixed up by native speakers. A semi-annual reading of this lesson will help more than a bi-annual one.

Bi-

Bi- comes from the Latin meaning two. When used with a temporal word, bi- means “every two” or “every other.”

This magazine is published bi-weekly, on the 1st and 15th of the month.

Congressional elections are held biennially, on even-numbered years.

The U.S. celebrated the bicentennial of its 1776 founding in 1976.

Semi-

Semi- comes from the Latin meaning half. When used with a temporal word, semi- means “twice.”

Our semi-annual meetings are in January and July.

I get paid semi-monthly, on the 5th and 19th of each month.

The semi-weekly deliveries occur on Monday and Thursday.

The Bottom Line

The confusion between bi- and semi- occurs because both prefixes are related to the concept of two. But bi- means something that happens every other (week, month…), or every two (weeks, months…), while semi- indicates something that happens twice every (week, month) or every half (week, month).