The way an ideal Zener diode is supposed to behave is, that it is non-conductive (in reverse bias) until a certain break-down voltage and above that, it is perfectly conductive.
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This is probably the circuit you meant by "Zener voltage regulation circuit". This is a basic linear voltage regulator. I see no point in using it for LED (you are supposed to regulate current through an LED, not voltage across it; so just use serial resistor), but let's explain how it works:
Example 1) The Zener diode has a break-down voltage of 3V, there is a 2V supply. No current flowing, because the voltage across Zener is too low.
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Example 2) The Zener diode has a break-down voltage of 3V, there is a 3V supply. The voltage is just enough to make the diode conductive, so some current is flowing.
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Example 3) The Zener diode has a break-down voltage of 3V, there is a 5V supply. The voltage is way higher than the threshold. 3 volts are still used to keep the diode conductive. Where goes the remaining 2V? In this ideal case, nowhere, so current would be infinite. With a real Zener diode, the current is limited by the internal resistance of the diode, but the current will likely overheat the diode.
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Example 4) The Zener diode has a break-down voltage of 3V, there is a 5V supply. The voltage is way higher than the threshold. 3 volts are still used to keep the diode conductive. Where goes the remaining 2V? It is "used" to push some current through the resistor that is connected in series with the diode.
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The resistor should have high enough resistance to limit the current through Zener, so that it won't overheat. But too high resistance will limit output current, if you connect load parallel to Zener, current draw of the load must not exceed current through the resistor at 2V (otherwise, the resistor would "consume" higher voltage than the 2V, which would mean lower voltage remaining for the load).
Now, if you want to model the Zener diode using basic components, it would look like this:
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The resistor is there to simulate behavior in the breakdown region (as you said in the second paragraph). Voltage source simulates the fact that there must be at least 3V across it to keep it in breakdown region (this model only works in the breakdown region, when voltage across it is higher than the 3 V; for lower voltage, it just behaves as open circuit).
Whe you suggest using only small resistor instead of Zener diode, you are completely missing the "voltage across the diode" part:
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Here, the "small" resistor is just a short circuit parallel to LED, so there is nearly no voltage across it.