Say person A concluded that when a king makes a statement to his people about X, the people will revolt. And person B concludes that if that king makes a statement to his people about X, the people will not revolt. Could persons A understand person B's logic if there conclusions are different?
Yes, person A could understand why person B reached a different conclusion.
Person A might learn that B's line of reasoning incorporated different premises. Here, person A might think that the content of issue X included subissues 1 and 2, but learns that person B believed that X included subissues 1, 2, and 3.
Or Person A might learn that B relied on the same facts, but that B gave the various facts a different emphasis. Here, persons A and B might have understood issue X in exactly the same way, but still disagreed on the level of importance that the kingdom's people assign to the issue.
Person A could understand these sources of disagreement without agreeing with B's conclusion.
This example is using the word "logic" in a colloquial way to mean something similar to "line of reasoning" as opposed to the technical definition used in philosophy, computer science, math, etc.
a (1) : a science that deals with the principles and criteria of validity of inference and demonstration : the science of the formal principles of reasoning, "a professor of logic", (2) : a branch or variety of logic, "modal logic", "Boolean logic" (3) : a branch of semiotics; especially : syntactics (4) : the formal principles of a branch of knowledge
b (1) : a particular mode of reasoning viewed as valid or faulty "She spent a long time explaining the situation, but he failed to see her logic." (2) : relevance, propriety "could not understand the logic of such an action"
Philosophy mostly deals with the definition (a) given above and this question concerns the definition (b). So, if person A tries various things, such as putting themselves in person B's shoes or having person B explain their reasoning, then perhaps person A could understand person B's logic. But again, this is a different use of the word "logic" than the use most common in philosophy. Consider the situation where person B has more information than person A. If that is the case, then person A might be drawn towards a different conclusion than person B. If person B explains their perspective and person A finds out the missing information then it would be completely possible for person A to understand person B's logic, even though they originally had different conclusions. But again, this is using the second definition of "logic", not the first.
Conflation is the situation wherein someone treats two distinct things as one thing due to them sharing some property. This happens a lot when words have a technical definition as well as a common colloquial use. In this case you should be careful not to conflate the first definition of "logic" with the second.
It depends what we mean by “understand the logic”.
If A and B both think rationally and have access to the same evidence, yet come to different conclusions it must be because they value the strength of the evidence not in the same way or find certain types of inferences to be more reliable.
Sure, the outline of B's reasoning can be understood by A – we can call this “understand the logic”. But A probably cannot fully comprehend why B judges some evidence or types of inference to be more or less persuasive than he does.
Though much can be made transparent to A, I doubt that the complexity of B's different cognitive style and the influence of his background can be fully grasped by A.