The original question was how do you convince a non scientist. Abstract
arguments about curvature in space time will most probably not work very
well. So whether you take an abstract approach, or use the approach that
one clock appears to run slower should be dictated by what the non scientist
already knows, and what they believe.
So I think the more down to earth arguments that Einstein used in his
original papers might be more useful. In other words look at how events
which appear to be simultaneous in one frame are not simultaneous in another
(the lightning bolts & train). Look at how a light clock is observed to be
running more slowly when it is moving with respect to you.
But you also need to get the object of the arguments to understand that they
often harbor ideas which are at odds with things you can measure such as
NTN3. One of my favorite misconceptions is the orientation of the new moon
at sunset which most people tend to say is uprignt, probably due to a famous
painting showing it that way.
It takes time for people to begin to understand that their view of "reality"
is conditioned by what they believe based on common experiences. Physicists
have constructed a different "reality" based on their experiments and
observations, and we should understand that this reality is only the current
paradigm and is subject to revision as needed.
A recent example that I just saw was a report on the effectiveness of
massage after exercise. They thought that massage would increase blood flow
and help the "healing" process. In reality it reduces blood flow. So is it
actually helpful beyond being a pleasurable experience? Medicine is full of
things that people thought were beneficial, and have now realized was bunk.
It has also proven that some old cheap remedies sometimes work remarkably
well, while modern ones do not.
As to whether you use the concept of time dilation, that is purely a matter
of opinion, and expediency. If it helps people understand, then use it.
Most people do not get to the abstract level of thinking, and only about 30%
make it to the formal operational level. Once one understands the
ramifications of this, then arguments over the correctness of
interpretations seem pointless. You use a model for the phenomenon which is
appropriate for the stage of current student thinking, not a model which is
hopelessly beyond them. Such a model should challenge them, but not too
much.
Perhaps there a McDermott tutorial that could be used for relativity?