I don't think there is any record that puts her staying with a friend of her mother in August, 1961.
She arrived at Mercer Island to stay with a high school classmate (Susan Blake) of her's (Stanley Ann's) prior to the time Ann had learned to change diapers which I believe fixes the visit on August 5 or 6. Observe that the birth on August 4 could well have occurred on August 3 US time and date.
There are also enrollment records (for Stanley at the University of Washington for fall quarter, 1961, however fall quarter in 1961 started in late September (or the first of October).
The visit with Susan Blake was a stopover on a trip by S. Ann somewhere else--Ms. Blake says to visit Obama Sr. at Harvard; however Obama Sr. didn't get to Harvard until one year later.
That's the record of which I am aware--and I think this element is very important.
My speculative analysis of what happened is that she went to Kenya to marry Obama Sr. in February 1961 because marriage in Hawaii would have been criminal bigamy. Obama's personal record places her in Kenya within 72 hours of when he was born; he says she came back to Hawaii. How did she do that and when? BOAC would not have let her on the airplane at that late date in her pregnancy and there is testimony from the two missionaries that is what happened.
Routing to the US from Kenya and East Africa was through Glaskow Scotland to Vancouver BC--I believe she went from Vancouver to Susan Blake's home on Mercer Island; then to Hawaii to file a resident's birth affidavit on August 8.
If you read the link which I provided back in September it distinctly says that Ann arrived at Susan Blake’s home driving the car of her mother’s friend. Maxine Box also said she visited Ann at the home of her mother’s friend where Ann and Jr were staying. We also know because of Susan Blake’s description of Jr as “very new and pink” that Jr was a newborn who hadn’t had the melamine show up yet. The actual schedule of when classes started at UW didn’t matter since Ann was registered in extension courses which she could start and finish at her leisure.