What are the modeling approaches in this cartoon? What are the modeling approaches depicted here? Can you name them and their prominent proponents or a landmark model? Is there an accepted superior approach? Who prefers which approach?

(From: http://www.stat.duke.edu/~mw/fineart.html)
 A: $\Theta$ = parameters; $X$ = data

Apriorius: form belief about $\theta$ without data
Pragmaticus: focus on the data
Frequentistus: classical statistics, focused on the likelihood, $\Pr(X | \Theta)$
Sapiens: consider joint distribution of $\Theta$ and $X$
Bayesianis: Bayesian statistics, putting prior on $\Theta$ and then calculating $\Pr(\Theta | X)$

There's no universally superior approach, though some may be universally inferior.  But of course the cartoon is a caricature, and if you think about it too deeply it won't be funny anymore.
Personally, I'm a pragmatist, though not in the sense in the cartoon.  I generally think of the world and the performance of statistical methods as a frequentist; I like the output of Bayesian methods but don't usually want to choose a prior.  I focus primarily on the scientific questions I'm trying to address and will use classical or Bayesian methods according to what seems to work best for the current problem.
I'll leave it to others to answer the rest of your questions.
