First, here are some things about the four generalizations you mention:
Monoids don't fall into Diers' framework: By his Proposition 1.4.1 the terminal object in his framework is strict, i.e. any morphism $1 \to A$ is an isomorphism, which is definitely not the case for monoids. I also wouldn't expect Diers' examples to be instances of Toen/Vaquie's framework in general, Diers' example 1.3.16, the category of pairs (ring, module over it), might be a counterexample. I don't know about Durov's setting.
Durov's geometry is in no obvious way an instance of Toen/Vaquie's framework. If you want to force it into that framework, this might be an idea to go after: Monads are monoids in the monoidal category of endofunctors. Commutative monads, however, are not commutative monoids in that category; indeed it doesn't even make sense to say that since the category is not symmetric monoidal. So first you have to find a symmetric monoidal ambient category in which Durov's generalized rings live. Seeing monads as Lawvere algebraic theories or as (things presented by) sketches might do the job - a commutative theory is probably exactly a sketch with an isomorphism from its tensor square. Another idea could be to consider a category of monads where morphisms are natural monad transformations together with distributive laws...
Derived algebraic geometry on the other hand is an instance of the homotopical version of Toen/Vaquie's framework, also contained in that article - see also below.
Second let me point out that there are many more generalizations of algebraic geometry than those four:
° Rings with extra structure can count as generalization, if one can endow any usual ring with such an extra structure, e.g.
Not Borger's geometry with lambda-rings: Not any ring can be endowed with the trivial lambda-ring structure .- see his comment
Berkovich's analytic geometry: Any ring can be endowed with the trivial metric
° One can replace rings by first order structures in several ways:
Several Russian authors do this in somewhat similar ways, a recent reference is this one by Daniyarova, Myasnikov, Remeslennikov which has many references to other work in this direction; see also this one by B. Plotkin.
First order structures can be described by sketches and there is an outline of algebraic geometry along this line in this text by R. Guitart.
° There are hyperrings (used for algebraic geometry by Connes/Consani) and fuzzy rings (by Walter Wenzel and Andreas Dress, e.g. this), which are certain second order structures.
° There is another generalization made by Shai Haran in his article on "dyslectic geometry" in which rings are endowed with gradings over general monoids (see here). Something quite similar seems to be going on in James Dolan's generalization of algebraic geometry (unpublished, but see here, there also was a series of videos of talks somewhere)
° derived algebraic geometry doesn't necessarily have to be based on simplicial rings; dg-algebras and $E_\infty$-ring spectra are equally important inputs, and there are many others, captured in
Toen-Vezzosi's HAG-contexts (these are homotopically additive)
Lurie's structured spaces from DAG 5 which capture about everything based on the idea of glueing together homotopical algebraic structure.
° note that Toen/Vaquie in their relative geometry do not stop at commutative monoids in some monoidal category but also give a homotopical version - this is something like a non-additive version of HAG-contexts and covers e.g. geometry over the "spectrum with one element" whose input are simplicial monoids.
° replacing rings by groupoid objects in rings together with an appropriate notion of equivalence gives you stack theory. Of course you can go on to higher stacks.
° of course there are the several approaches to non-commutative algebraic geometry - see Mahanta's survey for some of them, many related to the next point
° I am sure I forgot several things...
Third, since I am at it, let me note that there are also generalizations of algebraic geometry which do not exactly build upon a generalized notion of ring, e.g.
° Hrushovski/Zilber's Zariski Geometries (see here) capturing the essential structure which is used in the applications of model theory to arithmetic geometry
° Rosenberg's noncommutative geometry. It doesn't have to be non-commutative, and also not additive, as Z. Skoda pointed out here
° in particular: schemes as dg-categories (Kontsevich, Rosenberg, Tabuada)
To summarize: Carefully mapping AG-land will keep you busy for a while. I have gathered quite some material for a rudimentary map (or maybe a low resolution satellite photo) accompanied by a few selected closer snapshots, but I won't start writing it before the second half of the year...