Are there Interpretations of Classical Mechanics, Classical Electrodynamics, and/or Relativity (either Special or General) Just like Quantum Mechanics has many interpretations (Copenhagen, Many-Worlds etc.), do any other theories in Physics have multiple interpretations? For example, does Classical Mechanics, Classical Electrodynamics, Special Relativity, or General Relativity have multiple interpretations?
 A: As you mention: in the case of Quantum Mechanics there is widespread awareness of the existence of multiple interpretations.
In the case of General Relativity I am of the opinion that the following is the case: there are actually multiple interpretations in circulation, but there is little to no awareness of that. What that means in practice is that those who have learned General Relativity are under the impression that the particular interpretation they are committed to is the very foundation of General Relativity.
This raises a philosophical/phychological question: if there are multiple interpretations in circulation, but the general community is not aware of that, can we meaningfully assert that there are multiple interpretations in circulation?

On the subject of multiple interpretations of General Relativity:
There is the issue of Mach's principle.
(I need to get some things out of the way first. Mach's principle was not proposed by Ernst Mach. It was first proposed by Einstein. In that sense a better name for it would be Einstein's Mach's principle. Ernst Mach held to a very austere philosophy of science. (Confer for example Mach's position on the existence of atoms.) In terms of Mach's philosophy of science proposing such a thing as Einstein's Mach's principle is out of the question.)
In the years that Einstein developed the General theory Einstein was convinced that Einstein's Mach's principle was fundamental to the theory. The astronomer de Sitter found a solution to the Einstein equations that describes an empty space, since called de Sitter space. But according to the interpretation of Einstein's Mach's principle that Einstein was committed to such a solution should not exist. For a long time Einstein tried very hard to find a way to invalidate the de Sitter solution, but in the end he had to acknowledge it.
Historians of science describe that soon after that Einstein abandoned Einstein's Mach's principle, as exemplified by Einstein no longer mentioning it in his articles, a stark contrast with his emphasis in the years before. This was in the early 20's
But as we know: ideas have a huge inertia. The physics community should have followed Einstein in abandoning Einstein's Mach's principle, but it is my strong impression that many people are totally committed to Einstein's Mach's principle.
(I should add: there are multiple versions of Mach's principle in circulation, and those who endorse a particular version are aware of the differrent versions.)


On the general theory:
I have an exclusive preference for the interpretation endorsed by Johh Stachel, director of the center for Einstein Studies.
In the mid-1990's John Stachel started using the expression 'inertio-gravitational field'. The purpose of that name is to emphasize that the general theory should be thought of as a unification of two theories. Other historians of science have followed John Stachel in using that expression.
The writings of historians of science are a rich source of information; historicans of science study the development of ideas in the physics community.
With the name 'inertio-gravitational field' there is an intentional analogy with the name 'electro-magnetic field'. Electrostatic force and Magnetic force were thought of as distinct phenomena. Over time it became more and more evident that the two are interconnected. This culminated in Maxwell's theory of the electro-magnetic field.
In the case of the general theory: theory of gravity and theory of motion used to be thought of as distinct areas of theory. But there was always that strong suggestion of a connection in the form of the equivalence of inertial and gravitational mass.
Another way of saying 'theory of motion' is to say 'theory of inertia'. Newton's laws of motion describe the properties of inertia.
Einstein's general theory is a unification of theory of gravity and theory of inertia: the theory of the inertio-gravitational field.

Once I had learned the 'inertio-gravitational field' interpretation I was sold. I expect that over time it will become the dominant interpretation because it makes all the pieces of the puzzle fall into place.
A: What the interpretations of quantum mechanics seem to have in common is that they try in various ways to make it look more like classical mechanics, by translating quantum probabilities into classical probabilities, making the wave function a real objectively existing classical object, or postulating hidden mechanisms that make wavefunction collapse happen.
On that basis I'd say that the closest thing to interpretations of special and general relativity are models that make them look more like Newtonian mechanics, by breaking the unification of space and time. This includes the so-called Lorentz aether theory which supposes that everything is made of waves in the same medium that carries electromagnetic waves. This model is arguably Aristotelian since in addition to Newtonian absolute time, it also has a special state of rest.
I'd also include special relativity as typically taught to undergraduates, since it eschews four-vectors in favor of quasi-Newtonian three-vectors and scalars, and also encourages students to divide spacetime into space and time through concepts like "simultaneity", "length contraction" and so on.
In general relativity there are a lot of space+time pictures based on specific (noninertial) coordinate systems, such as the expanding-space interpretation of FLRW cosmology, the waterfall picture of black holes (from Gullstrand-Painlevé or other infalling coordinates), and the idea that time and space swap roles inside black holes (from Schwarzschild/Nordström/Kerr coordinates).
A: I'm writing a second answer to expand on the first answer, but here I'm covering a different topic.
I will present the case that for classical dynamics there are two interpretations in circulation, but there is a special reason why that bifurcation happened.
Presenting my case requires quite a lot of background. That background must be stated explicitly, otherwise the case I want to present doesn't have a sufficient foundation.


We have the unique situation that we have three conceptual frameworks used side by side: classical dynamics, special relativity, and general relativity. In all other cases when there was a revolution in physics the newer theory obsoleted the preceding theory. For instance, the dynamic theory of heat obsoleted the caloric theory of heat.
But relativistic effects become significant only in extreme circumstances. The conditions where relativistic physics is necessary are so rare that it would be absurd to regard classical dynamics as obsoleted.
That raises the issue: are there insights from relativistic physics that we should port back to classical dynamics? And if we do port back, can that give rise to a distinct interpretation of classical dynamics?
Let me first discuss the principle of relativity of inertial motion.
We have galilean relativity of inertial motion and SR relativity of inertial motion. The common concept is that of an equivalence class of inertial coordinate systems. What changed in the transition from classical dynamics to SR is that galilean transformations were replaced with the Lorentz transformations. As we know, Lorentz transformation simplifies to galilean transformation in the limit of slow relative velocity.
The importance of the principle of relativity of inertial motion cannot be overstated.
The example offered by Einstein in his 1905 article couldn't be more exemplary. The current induced in a coil when a coil and a magnet are moving relative to each other. As we all know: Einstein pointed out that in terms of the lorentzian treatment of electrodynamics there are multiple representations, each calculated differently. It can be represented as a case of motionless coil with the magnet moving, or with a coil moving and the magnet motionless. As we know: the two calculation end up predicting the same value for the magnitude of the induced current.
Einstein pointed out the acute tension: the formalism of electromagnetic theory of the time was using elements that are not inherent in the phenomena.
This is the core lesson of relativistic physics: one should always concentrate on the elements that are common to all members of the equivalence class of inertial coordinate systems because those are the elements that are inherent in the phenomena. Conversely: those elements that are frame dependent are definitely not inherent in the phenomena.


Now the issue: should concepts from the general theory of relavity be ported to classical dynamics, and if so, in which form?
For a background to that question I want to treat the case of the perihelion precession of Mercury as exemplary. The case of the perihelion precession of Mercury underlines the importance of the concept of inertial coordinate system. The relativistic precession of the orbit of Mercury with respect to the inertial coordinate system is 43 seconds of arc per century. The cause of the precession is the fact that the space surrounding the Sun is not Euclidean. This non-Euclidean character of curved spacetime is the key point. by contrast: in terms of newtonian dynamics the prediction is a perfectly repeating orbit, as newtonian dynamics assumes space is euclidean.
What would happen if you would represent the motion of Mercury using a coordinate system that rotates with that same 43 seconds of arc per year? In that coordinate system the orbit would not precess. Would that change the amount of spacetime curvature? Obviously not. The spacetime curvature is not frame dependent. The representation of it is frame dependent, but the causality of the physics taking place is not frame dependent.
In all: in GR we have the same thing as in all other areas of physics: causality is not frame dependent.
The importance of this cannot be overstated: causality is not frame dependent.


Next I go over the case of uniform circular motion, in terms of classical dynamics.
You have an object, restricted in its motion by a string, the string is attached to a central point, the object has a velocity; the object is in uniform circular motion.
As we know, the string exerts a centripetal force, causing centripetal acceleration, sustaining the circular motion. We assume the existence of inertia. (Anyone who declines to assume the existence of inertia deprives himself of the means to formulate theory of motion at all.)
Mathematically you have the option of representing the motion using a rotating coordinate system, you then use a coordinate system that rotates with the same angular velocity as the circular motion of the object.
As we know, the equation of motion then has an additional term, the centrifugal term. This centrifugal term has a factor $\Omega$ that represents the angular velocity of the rotating coorinate system with respect to the inertial coordinate system. As we know: the equation of motion for the rotating coordinate system is dependent on using the inertial coordinate system as reference of motion. That is: there is no escape from using the inertial coordinate system as reference of motion. You can use a rotating coordinate system allright, but that rotating coordinate system is dependent on the inertial coordinate system as reference of motion.


All of the above is the context I needed to put in place.
There is another interpretation of classical dynamics in circulation that deviates from the above. There is no short name for that interpretation, I will refer to it as the causality-is-frame-dependent interpretation
Proponents of the causality-is-frame-dependent interpretation insist on the following: in the inertial frame there is only the centripetal force, and in the rotating frame there is also a centrifugal force. The centrifugal force causes outward acceleration. So this is a point of view that asserts that the causality that should be attributed is dependent on the choice of frame of reference. Different frame of reference: different causality.
Proponents of the causality-is-frame-dependent interpretation tend to be combative, and insistent. They will insist that they are obviously right that that anyone who disagrees is obviously wrong. Proponents of the causality-is-frame-dependent interpretation see themselves at the leading edge of understanding physics and regard anyone who disagrees as simply not up to speed.
An example of such a combative proponent is the astronomer Phil Plait. Phil Plait has a website titled 'Bad astronomy'. The purpose of that website is to address cases of misunderstanding astronomy or physical science in general.
From time to time people point out to Phil Plait: "Mr. Plait, what you call centrifugal force is actually inertia." And Phil Plait's reaction to that is one of being annoyed, and he doubles down: "In the rotating frame there is a centrifugal force."


I'm not sure how proponents of the causality-is-frame-dependent interpretation arrive at that position. Clearly they are under the impression that they are porting ideas from the general theory of relativity back to classical dynamics. By the looks of it: proponents of the causality-is-frame-dependent interpretation feel they are asserting some form of relativity concept. What that relativity concept is supposed to be is unclear.


So we see a breakdown of communication. Proponents of the causality-is-frame-dependent interpretation will insist that what they believe is the only possible view, and that allowing any other possibility would be a regression.
A: They have different formulations (Newtonian/Lagrangian/Hamiltonian/...).
But because their central quantities are easily observable, these theories' interpretation is much easier than QM's.
A: Classical theories do not have multiple interpretations in the same way quantum Mechanics does. All theories require some sort of interpretation - i.e. a way to tie the mathematics back to the "real" world, however classical theories use concepts such as mass, position, time, EM fields about which there is little real argument (has not always been the case).
The concept of space-time introduced by Relativity is sufficiently similar to classical geometry that once again there is little real argument.
Note that interpretation of a theory is different from arguments about whether the theory is "correct".
Re interpretations: Quantum Mechanics is a completely different ball game; there are fundamental arguments over even what "elements of reality" the theory should be tied back to.
