Circuit Analysis – Average Power Dissipated as Heat in a Resistor

There is a well known, well studied phenomenon known as electromagnetic induction that is mathematically described by Faraday' Law of Induction.

Essentially, if the current through a loop of wire is changing, the associated magnetic field threading the loop is changing and, by Faraday's Law, there is an associated induced emf.

Now, to your question:

But what is happening in conductor so that the induced emf gets equal to E ? What is the picture of the mechanism ?

Let's assume the wire making up the loop is an ideal conductor or effectively so. Now, recall that there can be no electric field inside an ideal conductor*.

With the voltage source E connected to the loop of wire, the only way for there to be no electric field in the conductor is for there to be an induced electric field that precisely cancels the field due to the voltage source.

Since the requirement for an induced emf is a time changing magnetic field and associated time changing current, the time rate of change of current is precisely the value required to induce an emf that cancels the applied voltage.

*there are conditions in which this is not true but these are beyond the scope of this answer.

It's partly the precision of your measurements, and partly the effects of the meter you were using.

When you measured the voltage, you put the meter in parallel with the resistor and measured 4.94 volts, which is the value that your power supply is producing.

When you measured the current, you put the meter in series with the resistor, and measured 16 milliamps. However, the voltage across the resistor at this time was NOT exactly 4.94 volts, because the meter necessarily requires a voltage drop in order to measure current. This drop can be as much as 200 mV on most meters (at full scale), but in any case, it's proportional to the reading. In this case, your meter was probably causing a voltage drop of either 1.6 mV or 16 mV, depending on the measurement range you selected.

What this means is that the resistor was indeed dissipating slightly less power during the current measurement, because the voltage across it was slightly reduced. This explains why the calculation based on the current measurement alone is the lowest and the calculation based on the voltage measurement alone is the highest. The calculation based on the voltage and current together falls in the middle, because it is basically an average of the other two power levels.