Are scalar field theories considered metric theories?

Any advice on how Nordstrom's 1913 theory fails?

The Strong Equivalence Principle (SEP) has 3 components and Parametric
Post-Newtonian (PPN) formalism has 10 parameters. How do the 3
components of SEP relate to the 10 parameters of PPN?

Also, would I be right in saying that EEP and the WEP are essentially
the same?

I'm trying to write a short review of (4-D, nonquantum) alternatives to
GR, all the old theories that have since been rejected, and I'm trying
to figure out exactly how they fail. The fate of some theories has been
very difficult to track down, I've found one 2006 review that is fairly
complete in terms of theories but is packed full of mistakes.

Are scalar field theories considered metric theories? If so, the metric
is proportional to the Minkowski metric with the proportionality a
function of the scalar field.

Any advice on how Nordstrom's 1913 theory and Cartan's 1922 theory
fail?

Nordstrom's 1913 theory must not be confused with his 1912 theory. It
is the first consistent field theory of gravity that passes the WEP,
but fails the SEP (according to Pais). It's a scalar field theory with
constant speed of light but variable mass. It can't easily be
shoehorned into the description of general scalar field theories
discussed many years later.

Cartan's 1922,1923 theory is a non-metric theory of GR plus spin. MT&W
claims that Cartan's is the only non-metric theory that survives all
tests to that time i.e. 1973. The unreliable review claims that
Cartan's theory passes all experimental tests to 2006. Cliff Will
(1981) doesn't mention Cartan specifically but claims that all
non-metric tests fail. Will relaxes that stand in 2001 and gives EEP
test criteria for non-metric theories without mentioning Cartan.