Let K denote an inertial frame.
(Definition) Two events are simultaneous in K if and only if a symmetrically placed observer in K sees them, by the naked eye (through vacuum), happen simultaneously.
This definition is compatible with the classical conception of time.
Note: by saying "observer in K", it is meant that the observer is also at rest relative to K.
Instead of light, other signals could also be used, e.g. pistol bullets or even carrier-pigeons, as long as the symmetry of their propagation can be assumed. Since we are talking about mechanics, it seems reasonable to require only the symmetry of the net forces acting upon the chosen pair of (identical) "messengers". Einstein's original approach eliminates this requirement elegantly: it uses light signals and assumes that nothing whatsoever can make a difference to their propagation in vacuum. The same is definitely not true for e.g. a bullet whose motion is affected by a multitude of gravitational forces at the very least.
(Definition) Two clocks (at rest) in K are synchronized if the same positions of their hands are simultaneous events in K.