Quartz crystals: ESR and how to calculate power dissipation (or drive level)
The way I understand it is that the ESR includes R1 and also the
  resistive parts of L1 and C1. Is that true?


You've got too many "C1's" to answer without confusion. ESR reported by the crystal manufacturer is measured at series resonance, where ESR becomes very close to R1. In some test jigs, close enough to say R1 = ESR.
But keep in mind that most of these Pierce oscillators never run at the series resonant frequency. They run at a frequency above series resonance, and below parallel resonance (limited by C0). The motional model of a crystal is represented by L1, R1, in series with one of your C1. These are often shown as Rm, Cm, Lm. These component values vary with temperature, not with frequency. Remember that Rm, Cm, Lm are electrical models of mechanical resonance.


The whole feedback path (as seen by the inverting amplifier) has an equivalent resistance that does vary with frequency. At the oscillating frequency, its equivalent resistance is certainly higher than R1.

Since the crystal manufacturer evaluates the crystal at series resonance, assume that power limitation involves R1 (or Rm). A 100 uW crystal, with Rm=30 ohms can withstand 1.8 mA rms current through that series RLC branch. So you're on the right track, focussing on current and R1. At the actual oscillator frequency, this current can't be measured, only calculated from a model.