"at the beginning" would be more pointed. It would typically refer to the very start, i.e. September 1st (maybe the 2nd or 3rd) "in the beginning" is slightly more generalized. It would commonly refer to maybe the first week or so of September. While native speakers may use these interchangeably as suggested by other commenters, they do, in fact, have different meanings.

I would always say AT the beginning of August. However, we do say 'IN the beginning' in many contexts. So our brains get used to the preposition IN paired with 'the beginning'. So, although, personally, I would say AT the beginning of August/March/the year/the month I think others would sometimes use IN and not lose any sleep. I use IN the beginning in different contexts. "In the beginning, I had no idea how to use a computer" - I would also use AT in that sentence. AT does seem to suggest 'right at the start' whereas IN does, indeed, seem a little more fluid as to timescale. I think!

"In" is the better choice in this narrative. She is describing an interval on the timeline, not a specific instant of time. She is recording IN that interval.

Both 'in' or 'at' can be used. There might be some slight differences between the two, but it's minimal. I would say 'at the beginning of August,' personally.

The correct preposition in this case is "at", as in "at the beginning of August." We generally use "at" with specific points in time, like "at the beginning of," "at noon," or "at midnight." The phrase "in the beginning" is used more often in a broader or metaphorical sense, such as "In the beginning, humans lived in caves." So, for talking about a specific time like the start of a particular month, "at the beginning of August" is more appropriate.