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I think that you are reading the rule wrong. The first part states...
The referee may correct an obvious mistake by the timer to start or stop the clock properly only when he/she has definite information relative to the time involved.
You have definite knowledge if you see time still on the clock. Period. Thus, you are able to put time back on the clock.
The exact time observed by the official may be placed on the clock.
This part of the rule does not say that ONLY the exact time can be put back on the clock. It states that the exact time MAY be put back on the clock. Why? Because the rule in years past was that you had to allow for lag time. They have since changed that rule to allow officials to observe the EXACT time and put that back on.
Think about another situation -- if you have 30 seconds left in a game and inbound the ball in the backcourt. You get to a count of 5 and they still have not started the clock, so you stop the game, have the timer reset the clock to 25 and inbounds nearest to where the ball was when you blew your whistle.
That is definite knowledge but it is not "exact" -- but it is a perfectly legitimate situation under the rules.
The first statement of this rule -- that an official can correct an obvious mistake by the timer as long as the official has definitely knowledge, is independent of the second sentence that the exact time may be put back up.
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