David,
It's a good attempt. Allow me to throw in some corrections below.
dwoz wrote on Sun, 16 May 2004 18:35 
when an analog signal is sampled, it is essentially 'sliced' into segments, each of which has a certain width in time.

No. The samples have no "width" in time at all. They are momentary and instantaneous samples of the amplitude of the waveform at moments in time and do not imply what comes before or after them. Only on a computer screen are we led to think that the samples have width, because a computer screen tries to draw it up there in a way that is pleasing to the eye. In reality, sample points should be given to us as mere dots in space with no connecting line between them at all. That would help drive home the point that the samples are infinitely short in time.
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These 'slices' have to be made small enough that the signal being sampled cannot change direction TWICE inside the slice. (that's the nyquist thingy). Another way of saying this is that the frequency of the signal must be limited so it CAN'T go south then north inside a single slice.

More complicated than just going north and then south between two samples. Higher frequency content doesn't always manifest itself like that. Higher frequency content also causes steep angles and corners and much more. All of those behaviors are illegal also. Trying to adhere to Nyquist by means of discussing movement between samples implies a degree of misunderstanding about the genius of Nyquist and the role of reconstruction filters.
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OK, so when we check the voltage of this sample, we really have to make TWO measurements...the absolute level, and also the slope of the wave. without these two, we don't really know the complete behavior of that signal. But the measure we take is only the level. BUT, because of the definition we've given, that the signal will only go up, or go down, or go up and down ONCE within the sample, we can check the samples on either side and infer a slope for the width of the sample. It isn't perfect, but its pretty much good enough for our purposes.

No no no, and this is the dangerous part that people are led to believe. This is also the beauty of Nyquist. Nyquist came forth with a radical theory that you can know only ONE piece of information in order to completely accurately reconstruct the waveform, and that one piece is the amplitude at regular intervals of time. With a little trick math the ENTIRE rest of the waveform can be 100% of the waveform can be entirely and accurately reconstructed with proper amplitude, frequencies, and phase. In other words, we don't need to know the slope!! If we know only the amplitude of regular intervals Nyquist tells us that we can
figure out the slope. You must admit that this is a pretty revolutionary concept  one that defies common logic and simple intuition. And THAT is why it took over 20 years before a mathematician proved Nyquist correct.
So the point is that digital audio is not at all imperfect as you imply because we have all the information needed to reconstruct the waveform.
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What we're talking about here is QUANTIZATION ERROR. by knowing both the absolute value of the sample (either the leading edge, the exact middle, or the tail edge) and the interpolated slope, we can make some pretty good guesses about what "number" we should put in the output stream, so our DAC can reconstruct an analog signal that is very much like our input.

I'm not following this yet but I don't believe you're on the right path? Quantization error is only the error yielded by having to round the sample up or down to the latest quantization step when we take the sample.
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So, the sample rate controls HOW WIDE (or long) that sample is in time.

Semantical correction  it controls HOW OFTEN the samples are taken, not how wide the samples are.
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The bit depth controls how fine the discrete measurements we can make can be accurately captured. If we're trying to represent the whole dynamic range of the signal with just 8 bits, then we have to make some pretty gross approximations of the value of the particular we just measured. If we're using 28 bits, then the chance that the EXACT value of the sample that we just measured is only a tiny bit off a discrete integer that we can record, is much better.

Right!
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It should be obvious that BOTH the sample rate, and the bit rate, contribute to the QUANTIZATION ERROR.

Absolutely not!!! Only the bit depth relates to quantization error. Sample rate is completely unrelated.
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If I have a very high sample rate, then the more likely that my measurement of the slope is close to the actual value. I can rely on the absolute value of the sample, and less on the slope...and the higher the bit depth, the closer the measured value will be to one I can record. Remember, I have to round any 'inbetween' floating point value to an integer.

no, no, no, as I explained above.
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So, resolution of digital signals is dependent on both the sample rate, and the bit depth, to some extent.

Errr...language like this is specious and not the most descriptive way to describe what is happening.
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In the context of what I've just said, the "resolution" of analog signals would be to all intents and purposes, infinite. Not REALLY infinite of course, and when I can count individual electrons and quanta, then we can enter that discussion of just HOW INFINITE it is.

Again, not the best way to describe, for it denies the reduction in accuracy on the analog signal due to added noise and distortion. Since digital signals add less noise and distortion it is hardly accurate to say that analog signals have more "resolution," and is one of the reasons why it is best to shy away from that term  it does not do an accurate job of communicating anything pertinent about waveforms. While one can look at dictionary definitions and find that the term is theoretically relevant, it is actually quite inappropriate and doesn't adequately describe useful information in the context of analog vs. digital recordings.
Nika.