I have always understood that exact and precise are not synonyms although most dictionaries use one to define the other and sometime list them as synonyms. Please see the following example questions and answers that illustrate how the two words differ for me - "How many beans are in the jar? There are exactly 453." "How much acid is in that beaker? Precisely 13 ounces." The beans can be counted so you can state that there are exactly 453. The acid must be measured so it can only be precisely 13 ounces because the accuracy of the measurement is dependent on the precision of the measuring tool and the method of measurement.
If you measure a stack of paper with a ruler and say it is exactly three and an eighth inches tall, how can you explain that when you measure the same stack of paper with a machinist's micrometer and find that it is 3.127 inches tall. Even the micrometer does not provide an exact measurement. Some scientific measuring tool may be able to measure the stack to 10 decimal places. To my ears misusing these tow words sounds as awkward as asking "How much people came to your party." or "How many water can you fit in that jug?" I know I am too sensitive to the use of these tow words, but is there any basis for the distinction I hold so dear?
I think the quick answer is that for non-technical purposes, the words mean the same thing.
In my field, we sometimes perform Measurement System Analysis (MSA). In MSA, we analyze the measurement system by three related categories: resolution, precision, and accuracy. Resolution reflects the smallest unit that can be measured by the available measurement system. Precision means a very specific thing which breaks out into two factors: repeatability and reproducibility. Accuracy breaks out into three factors: bias, stability, and linearity.
It sounds like your personal definition of precision gets mixed in with resolution.
I am a bit confused by your example with a stack of paper. Are you saying that you object to "It is exactly three and an eighth inches tall" but you are OK with "It is precisely three and an eighth inches tall"?
Maybe the distinction comes in when the precision (or exactness) exceeds what is required for the purpose, like the report from the recent Norway bombing that a witness was standing "approximately 109 yards away" from the blast.
Or the time for the church service announced on this board:
I love it.
That is the point of MSA's focus on resolution. You want the resolution to be appropriate to the variation (or the desired, tolerable variation) in the process, around 10-20 divisions within the normal distribution. Lower resolution, and you can't see significant variation; higher resolution and you get a lot of "noise."
It seems like Wes bases the use of exact or precise on the discreet vs. continuous quality of the thing being measured. This would be something like the proscribed uses of few vs. less:
fewer than 10 items;
less than 10 ounces.
However, I have never heard of any such grammatical distinction for the words exact vs. precise. But for Wes, you can use exact only where some say we should use few, and you can use precise only where some say we should use less.
I can answer this from the physics point of view. This all comes under the heading of "uncertainty" wherein the terms "exact" and "precise" have scientific definitions, in use for about as long as science.
To say that a device or measurement is "precise" means that it delivers more significant figures (digits in a measurement for which the values are certain). For example, if you use a caliper to measure the thickness of a dime, you might get 0.14 cm. If you use a micrometer you could get 0.1355 cm. The micrometer is the more precise device, and it delivers more precise measurements.
"Exact" to my mind implies "accurate," and here again we have a definition. The "accuracy" of device or measurement is a measure of its error (compared to the true value), usually expressed as a percent. If you measure the speed of sound as 350 m/s, when the actual value is 340 m/s, your accuracy is 3%. I suppose you could also say your "exactness" is 3%. You could also say your error is 3%.
A high quality caliper might be rated as "accurate to 0.1%" and a cheap plastic one as "accurate to 1%" (even if they have the same precision).
One would expect that, in general, higher precision devices are also higher accuracy. Usually they are. But with a shoddy manufacturing run, bad design, or physical malfunction, the opposite can be true.