by the time establishes an end-point or end-range. It is used in constructions where the speaker wishes to say that something had happened (or will have happened) not later than the time specified. You can think of it as "<= the specified time", i.e. earlier than or equal to the specified time.
This traffic is terrible. By the time we get to the theater, the movie will be halfway over.
By the time the parents got home, the baby was already asleep.
By age three, she could recite the first 100 decimal points of π.
The lack of specificity you refer to in your question has nothing to do with the preposition by. All of the vagueness is in the time-phrase. In "by age three", age three really means something like "as a three-year old". She might have been 3-1/2, or even approaching the age of four.
I want you to be here by 10AM sharp.
There is no vagueness there. You should be here not later than 10:00:00.
By the time the clock struck twelve, the coach had turned into a
The coach was fully a pumpkin when then last chime sounded, or possibly when the first chime sounded. It depends on how you understand "struck twelve".