What should be the output resistance of this source with series diode?
If you know the v-i curve of the diode, you can take the slope of the curve at any particular point and call it the incremental resistance. This, for instance, is the norm when designing with zener diodes (if you want to do it right). It allows you to calculate the output voltage for small(ish) variations of load. Keep in mind that zeners tend to have a much more linear response than signal diodes when operated above their voltage "knee".

However, since the incremental resistance varies with voltage/current, for regular diodes it is only valid for a small zone centered around the measurement point. So, "output resistance" in the simplest sense is not a valid concept for figure 2. The output resistance will vary depending on what is connected to the circuit.