Yes I've added a figure to illustrate the meaning of $r(x), M_x, m(x)$.

I think zeta $\zeta$ specifically enters via Langlands computation of the integral of the Euler-Poincare form $\omega$ over the fundamental domain, but that is a paper i never could understand. "R. P. LANGLANDS, The volume of the fundamental domain for some arithmetical subgroups of Cheualley groups (Proc. of Symp. Math., Amer. Math. Soc., Providence, 1966. p. 143-148)."

Wolpert (1983) is interesting, especially pp. 252. But I dont follow your argument that $6g-5$ is sharp. My problem with length functions is that they dont readily distinguish between left and right Nielsen twists. And the lengths are like convex functions in the Nielsen twist parameter, and non monotone.

@AndyPutman yes i see your point. Another issue is that lengths are positive $\ell_a(g)>0$, so if you Nielsen twist around the red curves in the above figure, then the lengths of the green curves cannot distinguish betwen left and right Nielsen twists.

@AndyPutman Yes I agree that the sum of lengths of a collection of curves which fill the surface is proper. So sums of length functions do not provide coordinates on Teich following Agol's argument. But the possibility of length functions being coordinates still remains open from my POV.

@TimothyBudd So if we fix the orientation of the surface, then we distinguish between $S$ and its "mirror" $S^{op}$. But do you think it's possible that there exists other hyperbolic structures on $S$ where the six curves have a prescribed length?

Thanks for precise references. The eq (5.12) is quite elaborate. Buser's "Spectra", section 3.3 has further details.

@AndyPutman. I dont think Prof Agol's answer in the above question is correct, despite it's being accepted by the OP. For example the claim that "If one has 6g−6 geodesics which parameterize, then they [the curves] must be filling" is not accurate. Rather it is the collection of curves which needs be filling, and not the individual curves themselves. Moreover lengths of curves $\ell_a$ are nowhere minimized in Teichmueller space, they are only minimized "at infinity". That is length functions are not proper on Teich (their sublevels are like huge open horoballs).