Special Relativity Time problem

Question

Suppose there are two observers, they each own a light clock which were synchronised in the beginning and are moving relative to each other. So each of them accuses that the other person's clock is running slow. Now they see a concrete event, like someone clapping or a bulb shining and note down the reading on their clock and send each other the information, now if the two pieces of information were different, which means one of the clock was actually running slow, so they will know who was moving actually.

If the readings are same, then how will you justify Lorentz-Transformations for time ?

Answer 1

When both observers have the same distance to the event, then both will register the event at the same time with respect to their own clocks.

If you manage to keep the event source the same distance to the two observer for a period of time, you can even send a pulse train of events, and again both observers will register the events separated by the same delays.

None of this has much to do with Lorentz-Transformations.

However, when each observer fires another event whenever he registers an event, these events will appear time dilated to the respective other observer.

This however, does not have much to do with the external event source. You might just as well send any series of events which are equally spaced in time with respect to each observer's own clock.