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.