Can "among" and "amongst" have different contextual meanings?

After reading the answers to What is the distinction between "among" and "amongst"?, I realize there are no dictionary differences between the words and most would consider "amongst" a bit archaic. However, the following example seems to suggest there are at least contextual ambiguities that can arise if we use the words interchangeably.

Let’s discuss the issues among the departments.

Let’s discuss the issues amongst the departments.

The first sentence using among means there are issues between departments. The second sentence using amongst means we want to talk to the departments about the issues. - source: from the comments

Is this explanation correct, or are the proposed differences presenting a false sense of clarity where "issues" remains ambiguous regardless of using "among" or "amongst?"

Your boxed example deliberately presents an ambiguous sentence so as to let any biases from the word have maximal impact, but just because readers will experience some impact doesn't mean there is a "correct" difference between the words. When I read your boxed example, I understood the meanings in exactly the reverse of the way the box claims the meanings to be. One of the commenters had a third understanding. This shows that individuals have different inclinations regarding the nuances of "among" vs. "amongst". Since dictionaries also do not provide any distinction, it would be a mistake to think that some particular distinction is correct.

Every individual learns words through examples. People's understandings will lean towards whatever specific examples they have seen of its use, and for words like "amongst" which are not so common, there will be a random bias, varying from individual to individual, due to the specific examples they happen to have seen. As people see more examples, their understandings converge to a common understanding. (This is related to "the law of large numbers.")

The intent of a dictionary is to shortcut the process of seeing thousands of examples, by trying to directly convey the understanding that all these examples would give you. Not surprisingly, the dictionaries are giving you good information regarding "among" and "amongst": there is not any widespread reliable difference in understanding between these two words.