What makes an equation an 'equation of motion'?
An equation of motion is a (system of) equation for the basic observables of a system involving a time derivative, for which some initial-value problem is well-posed.
Thus a continuity equation is normally not an equation of motion, though it can be part of one, if currents are basic fields.
In general, a dynamical equation of motion or evolution equation is a (hyperbolic) second order in time differential equation. They determine the evolution of the system.
$\partial_{\mu}F^{i\mu}$ is a dynamical equation.
However, a constraint is a condition that must be verified at every time and, in particular, the initial conditions have to verify the constraints. Since equations of motion are of order two in time, constraints have to be at most order one.
The Gauss law $\partial_{\mu}F^{0\mu}$ is a constraint because it only involves a first derivative in time in configuration space, i.e., when $\bf E$ it is expressed in function of $A_0$ and $\bf A$. Furthermore, the gauss law is the generator of gauge transformations. In the quantum theory, only states which are annihilated by the gauss law are physical states.
Both dynamical equations and constraints may be called equations of motion or Euler-Lagrange equations of a given action functional. Or, one may keep the term equation of motion for dynamical equations. It is a matter of semantic. The important distinction is between constraints and evolution equations.
Conservation laws follow mainly from symmetries and from Noether theorem. Often but not always, equations of motion follow from conservation laws. Whether one considerers one more fundamental is a matter of personal taste.
Dirac equation relates several components of a Dirac spinor. Each component verifies the Klein-Gordon equation which is an evolution equation of order two.
OP wrote(v2):
What makes an equation an 'equation of motion'?
As David Zaslavsky mentions in a comment, in full generality, there isn't an exact definition. Loosely speaking, equations of motion are evolution equations, with which the dynamical variables' future (and past) behavior can be determined.
However, if a theory has an action principle, then there exists a precedent within the physics community, see e.g. Ref. 1. Then only the Euler-Lagrange equations are traditionally referred to as 'equations of motion', whether or not they are dynamical equations (i.e. contain time derivatives) or constraints (i.e. do not contain time derivatives).
References:
- M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.