In both cases, they annualize and then do the math, rather than do the math and then annualize. This calls to mind the associative property of mathematics, which says the order does not matter. While in addition, this holds (2 + 4 = 4 + 2 = 6, for example), it does not in these methods, as we get different results.
This also reminded me of how some attempted to derive my favorite risk-adjusted measure, Modigliani-Modigliani (annualize first or last?).
The information ratio differences:
And, the differences with Sharpe ratio:
This raises numerous questions. For example, is one approach superior to another? My suspicion is that most firms use what I refer to as the "typical" formulas. The reference materials I've checked only show these approaches. But clearly, some favor the alternative. The CFA Institute's CIPM(R) program, as I recall, also references the "typical" approach.
We know that there are multiple ways to derive various statistics, and here are just two more cases. As I learn more, I will share it with you. In the mean time, feel free to "chime in" with your thoughts.