If an online payment site asks me to enter the expiry date by selecting the Month name, they are asking me to perform a mapping from the numerical information on my credit card to the desired format. This adds unnecessary cognitive load. On the other hand, if the month format on the web page is numeric, then I can just rely on my working memory to transfer from card to screen - no mental processing required.
The example on the left is interesting. They've had a bet both ways. However, they introduce a new problem. The "Mar 03" format matches with another pattern in our minds (suggesting March 2003), thus potentially confounding the user, rather than helping them.
There are so many little things we can do when we design interactive systems to make the experience natural and - lets face it - easy for people. There are so many accidental errors we can make, that I am always surprised to find examples where someone has made a conscious design decision which adds no functionality, but increases users' cognitive load.