What is the meaning of the nominal frequency specified by manufacturers of crystals?
I think your second app note from ECS explains it: it is the frequency the crystal oscillates at when put into an (parallel) oscillator circuit, with the load capacitance specified by the manufacturer. (see page 85, "load capacitance")

This means you are right in that the frequency depends on the load capacitance - which is why it is specified in the data sheets.

But there are also crystals which are used in series oscillator, the you have no capacitance, and the specified crystal frequency is the series resonant frequency (fs).