What actually is the load capacitance of a quartz crystal?
Quartz crystals do not operate in vacuum in real world. They are usually connected to an amplifier in a feedback, so they are "loaded" with input capacitance and output capacitance. These capacitances cannot be avoided. However, if you add these caps, the resonance frequency (more specifically, a frequency point where the provided feedback turns 180 degrees at gain 1, or whatever) will be different from formulas you listed.

Therefore, the actual frequency cannot be specified without knowing the value of load. And this load depends on PCB design (traces/pads), and pin capacitance of different ICs. So, instead of making crystals for unspecified "thin air", manufacturers make the crystals to certain pre-defined load, say, "12 pF". The task of product design engineer is to meet this load with external capacitors, the value of which must be reduced for particular board parasitics and for particular pin capacitance of the IC in use.