In Oxford's definition, "after a particular length of time" can be thought of as "after a particular length of time has elapsed."
So:
He will return in a few days time.
basically means:
He will return after a few days have gone by.
As for:
He could learn a song in about five minutes.
a good substitution preposition for this would be within:
He could learn a song within five minutes.
Another way we could convey this is:
It took him no longer than five minutes to learn a song.
As for during, think of that as "at some point in time during a specified interval".
Therefore:
My daughter was born in January.
means:
My daughter was born on some day in January.
And:
Hurricane Katrina happened in 2005.
means:
Hurricane Katrina happened on some day during 2005.
Bottom Line: in cases where in means during, that doesn't mean within some interval of time (such as in five minutes, or in two days), but rather for a more specific range of time (such as in the first week of the month, in December, or in the 18th century, or even in the early hours of the morning).
Best Answer
She could say:
I was born in {city name}, in the state| province of {name} in {country name}.
She could go on to say:
But my family were from { village name | town name }, a small town|village about a three-hour drive from {city name}. There was no hospital in { town|village name}.