Disagreements of Analytical Philosophy and the Philosophy of Language
I'm going to respond from the position of someone who is clueless enough about the philosophy of language to try to relate the state affairs. You are keen to note that philosophy can be viewed as the process of converting intuition to language. In the analytical tradition, the literal process is subject to the philosophy of language which seeks to understand the relationship among concepts such as thought, reason, language, and theory, and otherwise build upon our understanding of the world derived from our naive realism. This can be conducted by an individual fluent in natural language with no formal training in the broadest sense, the sense that pre-Socratics did under basic labels of the physiki and theologi or it can be conducted with sophisticated artificial languages and formal systems which tends to exercised by the professional philosopher and constitutes philosophy in the narrower sense.
Frauds and Crackpots (Disagreement of Legitimacy)
For C1, this is certainly a possibility. Noam Chomsky, who is a popular and influential philosopher who we can thank for the notion of generative grammar has called at least one of his Continental contemporaries a fraud and derides certain methodologies as nonsensical. In fact, the Sokal affair is literally empirical evidence that some of what passes for postmodernist philosophy might be nonsensical in some sense. The problem with adjudging fraud in the broader sense where a lie is a communication with the intent to deceive is the age-old problem of knowing another's mind. Certainly sorting crackpots from geniuses is a big ask, though some metrics have been offered when analyzing theories. In US history, the Indiana PI Bill shows what happens when questionable philosophy gets into the ears of the uninformed. So certainly, C1 is a valid recognition of how some discourse is conducted. These "philosophical" disagreements use language so loosely they abuse it.
Differences of Opinion and Interpretation (Disagreement of Coherence)
C2 exposes another common happening in discourse. As the act of definition and establishing identity are both in the domain of philosophical methods, a lot of philosophical wrangling occurs over who said what. In my own readings, I've seen a lot of mischaracterizations of John Searle's arguments such as what exactly constitutes the social construction of reality and the Chinese room. One criticism I read of his views on constructing reality claimed he was promoting post-modernism which is absolutely naive interpretation of his theory given his analytical philosophy of mind. Daniel Dennett frequently and amusingly responds to criticisms that attack his positions that aren't his positions, and Thomas Kuhn remarked he himself was not a Kuhnian thinker since the term has changed substantially from his own claims. This is why natural language is often reduced to formalisms, to help clarify. In a formal framework, the syntax is lifted to prominence and it becomes hard to disagree. These philosophical disagreements are often a manifestation of differences in first principles, definitions, and often underscore a lack of familiarity with another's positions.
Shibboleths and the Defeasibility of Human Reason (Disagreement of Means)
No matter how many times a certain class of philosopher who seeks to obtain certainty from language that cannot be had wrings from deductive methods conclusions, the fact is that human reason is defeasible. I personally see this as a form of sophistry, and an entire analytic school rebelled against the abuses of technicalities in the form of advocating ordinary language use. In their (and my) eyes, cooking up even more sophisticated uses of natural and artificial language often is tantamount to trying to obfuscate the common-sense intuitions of thinkers. There is nothing inherently wrong with adopting conventions of language that clarify, but if your language is so opaque that the intuitions that it recognizes aren't clear, then the language-game you are playing might be a fancy game of who-knows-the-shibboleth. The real challenge is that theories are very complicated linguistic constructs, at the bottom of language there is an inescapable normativity. There are some schools of linguistic thought that are looking to reduce the ambiguity, for instance, those who advocate a natural semantic metalanguage or present semantic theories that are broader than truth-conditions. And that's just the interpretation (semantics) and representation (syntax). What of the differences and uses of methods of inference such as deduction, induction, and abduction (and inference to best explanation if you don't treat the two synonymously). These philosophical disagreements might be considered tactical disagreements because they are differences in the mechanisms of language.
Philosophers Will Be Philosophers (Disagreement of Ends)
Lastly, we come to your C4, where you note that not all philosophers conduct themselves to the same ends. What exactly philosophy is has been a matter of contention among some philosophers. Some philosophers are absolutists of sorts who reject the validity of any school of thinking but their own. Other philosophers (myself included) believe that various schools of philosophy are in essence nothing more than different technology-practices with differing metaphysical presuppositions. The logical positivists tried (and failed by their own admission) to use logic and science to eliminate metaphysics altogether! Philosophical disagreements of this sort might be called strategic differences since the plan on how to get to the goal is substantially different.
Metaphilosophical Disagreement (Disagreement of Disagreement)
Lastly, leave it to philosophers to disagree about disagreement. The Stanford Encyclopedia of Philosophy has an article Disagreement. From it, you can see that one can elevate disagreement from ordinary language analysis to a game-theoretic examination of values and incentives using ontological primitives of epistemology. So, you missed a C5, which is a special case of philosophical disagreement: "meta-disagreement"!.
