  Dynamic networks are a complex subject. Not only do they inherit the
complexity of static networks (as a particular case); they are also sensitive
to definitional subtleties that are a frequent source of confusion and
incomparability of results in the literature.
  In this paper, we take a step back and examine three such aspects in more
details, exploring their impact in a systematic way; namely, whether the
temporal paths are required to be \emph{strict} (i.e., the times along a path
must increasing, not just be non-decreasing), whether the time labeling is
\emph{proper} (two adjacent edges cannot be present at the same time) and
whether the time labeling is \emph{simple} (an edge can have only one presence
time). In particular, we investigate how different combinations of these
features impact the expressivity of the graph in terms of reachability.
  Our results imply a hierarchy of expressivity for the resulting settings,
shedding light on the loss of generality that one is making when considering
either combination. Some settings are more general than expected; in
particular, proper temporal graphs turn out to be as expressive as general
temporal graphs where non-strict paths are allowed. Also, we show that the
simplest setting, that of \emph{happy} temporal graphs (i.e., both proper and
simple) remains expressive enough to emulate the reachability of general
temporal graphs in a certain (restricted but useful) sense. Furthermore, this
setting is advocated as a target of choice for proving negative results. We
illustrates this by strengthening two known results to happy graphs (namely,
the inexistence of sparse spanners, and the hardness of computing temporal
components). Overall, we hope that this article can be seen as a guide for
choosing between different settings of temporal graphs, while being aware of
the way these choices affect generality.
