The two scenarios appear to describe different loans, if you ignore the artificial designations of "fees" and "interest" and simply look at the actual cash flow.
In the first, you walk out of the lenders office with $200 to do with as you wish. One year later, you come back with $216, and everything is paid off. The payment in excess of the amount loaned ($16), as a ratio to the amount loaned ($200), is 8%
In the second, you are initially given $200, but before you get to the door, the lender reclaims his $6 fee, so you walk out the door with only $194. Still, when you come back in a year to pay off the loan, you need to pay back $210. The ratio, the same as calculated above, is $16/$194, or 8.247%
The real error is in the example cited; it's wrong. The payment of the fee is correctly converted into a reduction in the true amount loaned, but the repayment of the fee is not included in the end-of-loan repayment...