This wikipedia article claims that there are two interpretations of Nordstrom's scalar theory of gravity:
1) A scalar field theory on flat space. The reason why an apple falls is that its mass is coupled to $\phi$, which causes it to deviate from a geodesic.
2) A theory of curved space-time. $R = k T$. The metric of space-time is $g = \phi^2(x) \eta$. The falling apple travels on a geodesic.
My question is, why are these considered two interpretations of the same theory, and not two separate theories? Although both give the same predictions for the trajectories of a particles, shouldn't the time elapsed on a watch differ by a factor of $\sim \phi$ between the two theories?
In other words, suppose that a particle gets displaced a small amount $\delta x$ in space-time. The proper time elapsed on its watch is according to 1) is $\delta \tau^2 \sim \eta \delta x \delta x$, and in 2) isn't $\delta \tau^2 \sim \phi^2 \delta x \delta x$? On the other hand, wiki says:
which metric is the one which according to Nordström can be measured locally by physical experiments? The answer is: the curved spacetime is the physically observable one in this theory (as in all metric theories of gravitation); the flat background is a mere mathematical fiction