Equations of motion (EOM) are typically the equations that determines time-evolution of the system.

E.g. in Newtonian mechanics, Newton's 2nd law is the EOM. (One should avoid referring to the kinematic $suvat$-equations as EOM to avoid confusion.)

For Lagrangian systems, the Euler-Lagrange (EL) equations are referred to as EOM, even in non-dynamical cases like constraints.

In field theory (as opposed to point mechanics), EOMs are also called field equations.

Equations of motion (EOM) are typically the equations that determine the time-evolution of the system.

E.g. in Newtonian mechanics, Newton's 2nd law is the EOM. (One should avoid referring to the kinematic $suvat$-equations as EOM to avoid confusion.)

For Lagrangian systems, the Euler-Lagrange (EL) equations are referred to as EOM, even in case of EL equations for non-dynamical variables.

Let us now return to OP's title question. In field theory (as opposed to point mechanics), the EOMs are also called field equations.