Hi Joe,
I'm reading Encyclopaedia Britannica.
I still didn't exactly get the meaning of "at the same moment as", because I was confused by the sequence of the times.
For better undertstanding your explanation, I think firstly I should get the sequence clear. Is the sequence like this? --
(1) At 10a.m. of June 18, the first fighting happened.
(2) And at 5p.m. of June 18, Washington's choice (as commander in chief of the military forces of all the colonies) was made.
The word upon made me confused that Washington's choice might be made at 10a.m. while the first fighting happened at 5p.m., as well as Miller's "at".