Criteria on Outer Tables in SQL -- beware of this situation!

When using an OUTER JOIN, you should put criteria on the outer table in the join condition, not in the WHERE clause. However, I often see a "workaround" to avoid this simple and solid rule, which might seem to work but actually doesn't. Since it is hard to explain why over and over in forum posts, I thought it might be helpful to address that here once and for all with an example.