I think you're limiting yourself too strictly to the equations and missing their application. Suppose, not a causal model, but a causal question. My question is if I put a mass weighing X on a spring with spring constant k, what will the displacement be (after oscillations stop, under standard gravity g, blah blah)? That's a causal question, answered by hooke's law. You can construct analogous questions for all sorts of scenarios. I'd say a model can be considered causal if it can answer all (relevant) causal questions about what it describes.
Take something more dynamic, like a ball being thrown on a level surface (uniform gravity, ideal vacuum, etc). If I throw it with some force, it'll follow a perfect symmetric parabola and land with that exact same force. The force it lands with (y) is exactly identical to the force I threw it with (x). It's basically time symmetry, but simpler. The equation here is the symmetric y = x, so it won't help you define causation. But clearly the usual version of causation says that my cause is x and the effect is y.
Maybe where we differ is that you say it's "the causal model is assumed, then observations are made and a law is formed which can be used to make predictions," while I'd describe it as "interventions are made, then observations are made and a law is formed which can be used to make predictions (and the law itself doesn't depend on the intervention taken)." That's what makes the y = x scenario not symmetric. X is my intervention.
no physical law implies causation. most laws describe relationships between measurable quantities. there is no causation implied e.g. in e=mc2 or e=hv or maxwell's equations etc.
the causality is imposed post-hoc depending on your application, but it's not explained by the law itself. alternatively, causality is a prerequisite in order to have physical laws at all.
Take something more dynamic, like a ball being thrown on a level surface (uniform gravity, ideal vacuum, etc). If I throw it with some force, it'll follow a perfect symmetric parabola and land with that exact same force. The force it lands with (y) is exactly identical to the force I threw it with (x). It's basically time symmetry, but simpler. The equation here is the symmetric y = x, so it won't help you define causation. But clearly the usual version of causation says that my cause is x and the effect is y.
Maybe where we differ is that you say it's "the causal model is assumed, then observations are made and a law is formed which can be used to make predictions," while I'd describe it as "interventions are made, then observations are made and a law is formed which can be used to make predictions (and the law itself doesn't depend on the intervention taken)." That's what makes the y = x scenario not symmetric. X is my intervention.