The wording of the oath during the first inaugural was botched and redone the next day as a precaution, same oath.
The 2nd inaugural fell on a Sunday. We never have the public ceremony on a Sunday, so that happened on Monday. The oath delivered at that time was just for show, the one on Sunday is the one that counted.
Same oath. There is no silly conspiracy here.
Why do you believe it was the same oath? On the same Bible? (Or Koran, or whatever Obola picked.)