In 1) the output is Voltage on resistor "R", so it is referenced to Vdd. Your schematic shows it referenced to ground, which is incorrect. If Vdd is constant, both will be the same AC output with a different DC offset. However if Vdd is variable or noisy, referencing the output to Vdd will reject most of the noise, whereas referencing the output to GND will reject none of it.

Schematics 2) and 4) cannot work in practice because, as you noticed, source Vm sets the differential input voltage, but nothing sets the common mode input voltage. So you can split the Vm source in two identical sources Vm/2 like in schematic 3, and connect the middle point between these sources to the common mode voltage. In schematics 1 and 3 the common mode voltage is GND but you can use any voltage as long as it within the range allowed by your FETs. Note in schematics 1 and 3, both sources have value Vm, so differential input voltage is 2x higher in schematics 1 and 3 than in the others.

3. is like 1), but the output signal is taken from both drains and not from 1 drain and GND. This allows to double the gain.

yes, 2x the gain.

But, despite that difference, is there a reason to prefer 3) on 1) or viceversa? I'd say that 3) is better because noise on GND is not transmitted to the output signal. Is it true?

As said above in schematic 1 the output should be referenced to Vdd, not GND.

Schematic 3 provides a differential output, which is useful if you need a differential signal (also see below). Another option is to replace the resistors with a current mirror, which results in a push-pull current output.

I have seen that the current source is often replaced by a simple resistor. In this case, it is important for it to be very high in order not to have common mode gain.

Schematic 1 would have common mode gain, but schematic 3 would not, because the output is differential. Variation in Ibias causes common mode output voltage variation, but not differential output voltage variation. However varying Ibias does change the differential gain.

But, in general, which is the reason for choosing a current source between Source and GND, or an high resistor between Source and GND?

If you're considering an opamp input stage and the input common mode voltage spans almost the whole range between the supplies, if you want a reasonably constant current, there is no way to do it with a resistor. Also a current source will result in better common mode rejection and power supply noise rejection.

Sometimes this kind of circuit works with VDD and -VDD, so with dual voltage supply. Which is the reason for choosing it or a single supply?

One supply is cheaper than 2 supplies if you can get away with it.

In 1) the output is Voltage across resistor "R", so it is referenced to Vdd. Your schematic shows it referenced to ground, which is incorrect. If Vdd is constant, both will be the same AC output with a different DC offset. However if Vdd is variable or noisy, referencing the output to Vdd will reject most of the noise, whereas referencing the output to GND will reject none of it.

Schematics 2) and 4) cannot work in practice because, as you noticed, source Vm sets the differential input voltage, but nothing sets the common mode input voltage. So you can split the Vm source in two identical sources Vm/2 like in schematic 3, and connect the middle point between these sources to the common mode voltage. In schematics 1 and 3 the common mode voltage is GND but you can use any voltage as long as it within the range allowed by your FETs. Note in schematics 1 and 3, both sources have value Vm, so differential input voltage is 2x higher in schematics 1 and 3 than in the others.

3. is like 1), but the output signal is taken from both drains and not from 1 drain and GND. This allows to double the gain.

yes, 2x the gain.

But, despite that difference, is there a reason to prefer 3) on 1) or viceversa? I'd say that 3) is better because noise on GND is not transmitted to the output signal. Is it true?

As said above in schematic 1 the output should be referenced to Vdd, not GND.

Schematic 3 provides a differential output, which is useful if you need a differential signal (also see below). Another option is to replace the resistors with a current mirror, which results in a push-pull current output.

I have seen that the current source is often replaced by a simple resistor. In this case, it is important for it to be very high in order not to have common mode gain.

Schematic 1 would have common mode gain, but schematic 3 would not, because the output is differential. Variation in Ibias causes common mode output voltage variation, but not differential output voltage variation. However varying Ibias does change the differential gain.

But, in general, which is the reason for choosing a current source between Source and GND, or an high resistor between Source and GND?

If you're considering an opamp input stage and the input common mode voltage spans almost the whole range between the supplies, if you want a reasonably constant current, there is no way to do it with a resistor. Also a current source will result in better common mode rejection and power supply noise rejection.

Sometimes this kind of circuit works with VDD and -VDD, so with dual voltage supply. Which is the reason for choosing it or a single supply?

One supply is cheaper than 2 supplies if you can get away with it.