Synonyms of "specific" and "precise" include accurate, detailed, explicit, stringent or rigorous. A precise measurement is one made with well-maintained equipment and using the correct methods in a careful manner.
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What determines how precise a measurement is
It is precise if it is repeatable.
The measurement of the grams of water in a kilogram of air is called specific humidity.
An accurate but not precise measurement would be an approximate measurement. For example, in cooking or baking, the ingredients list might indicate one cup of a certain item. Scientifically, however, one cup, (or 8 ounces) would not be a precise measurement for testing purposes. Instead, a precisemeasurement would be required, especially when the results of a specific test need to be reproducible and the quantities are minute, as in micrograms, for example. Accuracy in measurement is of closeness to the actual or exact, but precision in measurement is closeness to the same spot each time it is taken.
Is as exact as possible