So far the classical conception of time was untouched. What we've solely gained is a rather restricted, but rock-solid way of measuring time.
(Definition) In an inertial frame K, the time read off from the synchronized clocks is called the coordinate time of K.
The definition of simultaneity in K can now be extended to include those events that happen directly to entities in motion:
(Definition) Two events are simultaneous in K if and only if they happen at the same coordinate time.
The coordinate time of an event can be read off from the clock in K that is momentarily co-located with the entity to which the event happens.
As long as there is only one inertial frame considered, time seems no different from that of classical mechanics.