Jaw Harp- I consider this to be an Enclosed Free Reed, Hornbostel and Sachs consider this to be a Lamellophone
Plate Free Reeds- accordion, harmonica, melodica, reed organs, sheng, all Enclosed Free Reeds, neither open or closed, and effected by either direction of airflow.

Warpable/flexable lamell reed- an unenclosed free reed that is made out of a strip of highly flexible material and vibrates in response to being dragged by parallel wind force, can be inside a tube.

Lamell Reeds/Mouthpiece Reeds- Clarinet or saxophone Reeds, where a lamell beats against a rigid material called the mouthpiece, forced closed by air pressure.
Drawn double lamell- two parallel lamells that beat together through the forward drag of air flowing parallel between them, could be considered unenclosed free reeds, uniquely work without any external tubing.

Invert Mouthpiece Reeds/ Invert Lamell Reeds- Theoretically it is possible to make a reversed mouthpiece where the reed is forced open by air pressure. In such case the reed would face into the tube of the instrument and would have to be inside of a windcap.

Double Reeds- Oboe, Shawm, Bassoon, these are two framed membranes with a small edge that is open. Is forced closed by air pressure, beat against one another only at the opening.
Crushed Double Reeds- Duduk, Guanzi, Hichiriki, two membranes that have a narrow slot opening that vibrate against one another in a linear fashion.
Crushed tubular Reeds- when a segment of tubing that is flexible has a clamped restriction produced in or near the center that vibrates from wind passing through the restriction, edges beat against one another.

Invert reed- Like lips playing into a brass instrument (these are invert because its two edges that are forced open by air pressure instead of forced closed like on a double reed)
Linear Invert Reed- examples include binder clips- the opening edge where vibration happens of the reed is extremely long and pitch can be generated from the internal cavity and by the length of the opening that vibrates (wider wings on the reed take longer amounts of time to cycle). This is very similar to the function of invert reeds like lips listed above but it has a different way of modulating pitch where the overall length of the reeds opening can determine pitch instead of the case of lips where the tension of the closed edge (a type of Regulation). In theory a Pressure closed version of this reed could be made if it were entirely encapsulated into a windcap and directed into a narrow tube (for resistance).

Bullroarer- Called a Unenclosed Free Reed, I find describing this as a reed dubious, as it either acting as a string mount for the rope (source of agitation for the string) or a idiophone of some sort.
Grass Blade- This is a Air activated ribbon (a type of string) that is mounted on both sides, vibrates when air drags against the ribbon causing it to have torsional vibration.
Membranophone- A double chamber where air passes from the outer chamber into the inner chamber which is a tube by forcing open a membrane that is under tension, membrane beats against the tube.

Summarized from Wikipedia and Encyclopedia Britannica-
reed “A thin strip of material that vibrates to produce a sound on a musical instrument” “a thin blade of material typically a reed or thin metal that vibrates”

This definition has a major issue in that is describes single reeds (lamell reeds like those of a clarinet or saxophone) as well as free reeds (like those of a harmonica) but inadequacy describes the behavior of double reeds (or any other class listed above) because the focus is on the shape of the material.

A reed is any solid material that vibrates in a stable and consistent manner in response to wind as the generating force of the oscillation. This is distinct from how flutes operate as the solid material the flute is made of is not required to vibrate for the instrument to operate, only the Air Column or Cavity Resonance is required.

For Reeds the following forces happen-
Wind –> Material that vibrates in response –> Sound is reinforced and directed by a resonating tube (air column with standing waves)

For flutes the process works as such-
Wind –> An edge that causes an oscillation –> Sound is generated in the Air Column with Standing Waves or in a Helmholtz Resonator with Cavity Resonance ^{1) 2)}

The use of the word reed should perhaps be phased out. I find it contentious to name a set of phenomena after a material (one of many) that it is made out of. This of course carries over to other classes such as brass for brass instruments and wood in woodwind instruments.

It seems there is a ongoing consensus that the lips are not reeds by virtue that they are attached to the body, and not to the instrument. I find this distinction to be interesting as it is true the lips are only in contact with the instrument while playing, but they are also the physical source of the vibration. Similar could be said about lips being essential to many other winds such as the clarinet in the context of pitch regulation (which they also serve as such in brass instruments). I would counter this idea by the fact that a sort of artificial lips can be made ³⁾ and used to play brass instruments (robots can in fact play brass instruments using silicone or latex ⁴⁾ lips⁵⁾), and of course these synthetic ones are reeds in a more obvious sense, and can even be attached to the instrument in a more permanent way.