There are two parts to a fork, the spring and the damper. Solo air, dual air and coil are all types of spring and absorb energy when the fork is compressed. The damper, which is the part of the suspension system that removes oscillation after the spring has absorbed energy and is returning, is generally a hydraulic system filled with oil. The Lyric has an oil damper as well as a coil or air spring.
A modern air spring generally has two chambers one positive and negative, the positive provides resistance against compression and the negative resistance against rebound as the spring returns to it's normal length.
Solo and Dual air are RockShox terms for their proprietary air spring system and relates to the way the shock is filled, but both are dual chamber. Until Solo air forks became available (2012 or 2013 forks) it was necessary to tune both chambers to get sag and rebound correct, now solo air is a single fill system that does this much more easily. Dual air (as opposed to dual position air) has now been replaced completely by solo air on the RockShox range.
Dual Position Air
Dual Position Air is another proprietary RockShox technology and is a
type of air spring that allows for travel to changed between two travel lengths through a lever on the crown. It acts in the same way
as a solo air spring but gives the rider greater choice on a ride.
A coil fork uses a mechanical spring in place of an air spring. A coil spring is simply that, a big metal spring.
The benefits of the two types are often debated. A dual chamber air spring fork is lighter than a coil spring but requires more tuning. A coil fork has a finite stroke length so the rebound is smoother at the end of the stroke and makes the fork feel more plush throughout the stroke.
It would be redundant to have both a air spring and coil spring in a fork, as both are springs and neither are dampers. The fork would bounce back and oscillate (bounce up and down like a spring until all energy was released). There have however been bikes with two stage suspension systems like the Elite9 which used dual rear shocks and could effectively line up a coil and air spring shock.