I think there were three or four words used by the Greeks for
   logic. Analytika being one. Before all of them, we had "nyaya"
   in the East (circa 1700-1800 BCE).

   I think the Chinese had something as well which was later named
   "analytics" by the West. Nowadays, analytics is a branch of
   philosophy particularly dealing with patterns and permutations
   because that was a large part of the original logic school.

   We typically identity the logic theory by principle and method,
   not the name. Even if Aristotle called it analytics, we can
   still recognize the syllogism as classical logic.

   "Pure logic" is just a contrived name for what has been called
   absolute logic, natural logic, and universal logic in the past,
   transcendental logic since the enlightenment, and recently
   "ultimate logic". Transcendental logic is a bit of a
   metaphysical theory (recently, quantum theory), so modern logic
   chiefly ignores it. There are factions. What I call the white
   faction (Hegel, Descartes, Hume, early Wittgenstein) died out a
   long time ago. They weren't shy about discussing the
   metaphysical nature of logic and reason. However, as the
   analytics and realists began to dominate the discipline, we see
   a repeat of the final days of the Christian Gnostics.
   Fortunately, no one was killed.

   Then we have the black faction (Russell, Frege) who won the war
   and is now the primary movement in logic theory sans any
   metaphysical notions. And of course, you have people like me in
   the gray faction (Whitehead, later Wittgenstein, Kant) who
   although aren't afraid to mention transcendental logic theory,
   typically tow the line because we know if we speak too loud
   about it, pseudologic will become as popular a thing as
   pseudoscience - and nobody wants the woo.

   The places where fields like geometry and math intersect with
   pure logic theory are what we call the "principles of logic"
   which are rules, axioms, and laws in logic for which
   alternatives do not exist or cannot exist under its most
   fundamental conditions. Laws like the law of identity and axioms
   of choice. There are maybe around a dozen primary principles in
   logic and another two dozen derivative principles that cannot be
   reasonably questioned in any context without changing the nature
   of natural law itself. Stuff like causality and dimension which
   cannot be questioned under normal conditions.

   It would be like trying to argue that the shortest distance
   between two points is not a straight line. Someone once
   suggested that a wormhole broke this rule, but between the
   entrance and exit of a wormhole is still a straight line or
   superimposed points connected by a zero length linear dimension.

   Every new system of logic has its own agenda - a use for which
   is was purposely designed. However, despite the customization,
   every system of logic MUST adhere to the same principles. The
   Universal Logic project is recently attempting to systematize
   this theory in an easily understandable and translatable format.
   But basically, even though not all logics rely on the exact same
   principles, the principles they do rely on must necessarily
   resolve themselves to the parent.

   For instance, Godel's incompleteness theorems adhere to the
   principles of logic, and validly so, but not EVERY principle of
   logic. And although it may be valid is a classical sense,
   whether it is valid in a many-valued, higher order logic sense
   has been called into question by many a logician and found
   somewhat wanting.