It may be worth noting that the DeTurck trick is nonlinear, amounting to a coupling with the harmonic map heat flow, and the Uhlenbeck trick is from a linear ODE, resulting in a one-parameter family of self-maps of the set of bases of a tangent space. So arguably one shouldn't expect any relation.

English translation is at St. Petersburg Math. J. 5 (1994), no. 1, 205–213, I can't find it on the web though

It is of course true that the second definition doesn't need a metric. But it is unusual and not great for the addition of a structure to modify or restrict the meaning of of a definition. Is there any typical language for both definitions which is compatible with one another?

Thanks, very interesting!

thanks for the article, I've added a comment about it to the question

I'm not sure. Supposing it does, I think my condition should be significantly more restrictive, since I assume the lengths of geodesics in Besse's manifolds aren't usually all multiples of the same number

Depending on the particular applications you're interested in you might want to look at Eberhard Hopf "The partial differential equation $u_t+uu_x=\mu u_{xx}$" Comm. Pure Appl. Math. 3 (1950), no. 3, 201-230, or S.N. Kruzkov "First order quasilinear equations in several independent variables," which are concerned with these kinds of questions

The Liouville theorem 5.4 in Slovak's article gives the conformal group by giving a list of generators. I believe makt would like a parametrization of the group, e.g. the isometry group is more precisely described as mappings "Ax+b" rather than "the group generated by rotations and translations"

That's fair. When I called it the Reilly formula I was thinking of the case of a closed manifold

I agree that the standard textbook material in math must be among the most reliable in the sciences. And I think that what you say applies to certain results and certain methods in modern practice, but that a majority of research does not directly become part of the ecosystem in that way. In part it's like Voevoedksy said, "A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail." I think such an instance is not naturally checked against by #2 and 3, although of course in certain cases it is.

I agree, but I don't see how that's helpful for the problem of a present-day mathematician understanding present-day mathematics research, which is what I understand the replication crisis to be about.