Linearized metric for GW emitting orbiting bodies

[RockyMarciano]

This link is originally from a NASA site, now copied into Wikipedia
https://commons.wikimedia.org/wiki/File:Wavy.gif
The vertical displacement illustrating strain is what I would call "artistic licence", I have used colour coding instead:

I don't see why you think that isn't doable or where the error lies in what I've done. The colour coding shows the magnitude of the strain moving out from the centre but also how it creates the illusion that the wave rotates with the binary.

As PeterDonis has already commented the math behind the NASA animation is different from the math in the wikipedia page. The animation is an idealization showing GWs as spherical waves near their source, while the solutions showed in the wiki page are plane wave solutions(even if the use of spherical coordinates and the 1/r factor may be misleading) at detection.
Now in principle there should be no problem with this two different approximations of a wave phenomenon, GWs follow closely the analogy with EM waves and EM waves are routinely idealized as plane waves even if spherical in principle and there are solutions for both in Maxwell's equations.

But there is a point where analogies break and unfortunately in GR there is a different situation with respect to gauge degrees of freedom from the Maxwellian or Minkowskian case, and while in the latter case the Lorenz gauge fixing is enough, in GR unphysical degrees of freedom still remain after Lorenz gauging that are not taken care of by a fixed bacground geometry like in the EM case.

The additional traceless gauging(TT) that is required to leave only the physical degrees of freedom at the detection region far from the GWs source prevents from the use of spherical waves solutions.
Now from the point of view of the generation of GWs at the source(which is based in full nonlinear GR-strong field regime) and their 1/r multidirectional propagation from a central compact source this situation creates an awkward disconnect with the detection region modeled in the weak field linearized regime.
For this reason I consider quite futile to intend a graphical representation that includes the source and the far region of detection that is truthful to the math of GWs, there simply is no spherical wave solution of the EFE under the gauge in which they are effectively equivalent to a wave equation and without unphysical degrees of freedom.
The quadrupole moment used in the calculations of the strain based on the stress-energy of the source is necessarily traceless and transverse also, so this TT gauging is a consistency condition concomitant to the accuracy of the quadrupole formula

 

[PeterDonis]

the amplitude refers to the strain but omitting the time variation

Ah, I see, yes, that terminology is sometimes used.

The NASA animated gif shows a displacement in the direction perpendicular to the orbital plane which I think represents the local value of stretching in the plane.

I'm not sure that's what it's really showing. I think that particular image is more "schematic" than anything else, it's intended to convey a general picture of "waves propagating from the source", but it is not intended to be an accurate depiction of the detailed structure of the waves.