What is the role of re-enforcing mesh in plastering, when is it required, and which mesh types are prefered?
The only place I seen that fiber mesh used in the Exterior Insulation Finish System (EIFS) system of stucco. It goes over the insulation board, folded over the edges and mudded in with a product for the EIFS. It is similar to thinset for tile work, but I believe it is a different product. After that is coated in, the finish layer is applied. It serves to toughen up the insulation board. This is quite an over simplification of the whole system, but it gets the idea across I hope. I have never seen this used in the 3 coat process for stucco, only for the EIFS system. I have seen the EIFS system used on cement board without the mesh over the whole surface since the joints only needed a narrow tape. In the comments I see tahwos uses it for plaster repair, and touched base about the other more narrow mesh. The mesh I was referring to earlier was like the material you have pictured that come in a 38"X 150' roll(s)

You have asked about plastic mesh too. The only place I have seen plastic mesh used, is a much heavier gauge for landscape and roadwork primarily for earth stabilization. There may be others in the group that may know more on that.

The metal lath is the only base material I have seen the 3 coat stucco being used. Also, back in the day, when plaster was in use or even towards the end of its popularity, the metal lath was used in plaster repair and bathroom shower stalls and some tub surrounds too whenever tile was going in place. Of course I have seen whole walls plastered with metal lath as the base, but only in one house over my 47 years in the trade.