1) The Airy disk refers to the central disk. When you consider both the central disk and the surrounding circles you should call it the Airy pattern.
But in my opinion it is just arbitraty taxonomy hence it is not so important as long as you understand what happens.
2) What is the difference between the Airy disk and the seeing disk?
In telescope optics, the Airy diffraction pattern is caused by the diffraction of light by the finite aperture of your instrument and it puts a fundamental limit on the spatial resolution of your telescope. This is called the diffraction limit.
But realize that when performing observations from the ground, the signal which is detected by your instrument has crossed the atmosphere. As random turbulence arise in the air, the electromagnetic wave carrying the information may be affected by the local variations of refractive index induced by turbulence. Hence you are not limited by diffraction only, but also by the following effect: as the wave front is distorted along the path between the source and the detector, the quality of what should be a diffraction pattern will turn into a blurred and twisted signal, that you are calling the seeing disk.
Have a look at these images that illustrate the effect of turbulence:
Seeing really is what alter the Airy pattern and reduce the spatial resolution of your telescope.
When seeing is perfect (when there is virtually no turbulence), your are diffraction limited: your resolution is optimal and constrained by the aperture of your instrument. As the quality of seeing decreases (when the atmosphere is windy for example), the resolution is diminished and the Airy pattern becomes less visible.