What i am objecting to is that you use at least two different scenarios to prove a general point, and the two scenarios are not identical in nature.
I hope to make you understand that if you want to create an a priori rule that applies to ALL scenarios, then your premises can't be exchanged between two paths of reasoning, in a way that the one currently in use is the only premise that applies to the situation and the other premis currently not in use is forbidden to apply to the situation. Yet you do that.
What I read is not that you use separate specific cases, with the same rules, and with the same mechanism of logical constructs to arrive at a conclusion; but instead you use separate specific cases each with their own specific and non-overlapping different rules. And you can't, must not, unify these rules, because the mechanisms you apply in the different cases are also different; yet you claim that your unifying the rules are valid.
More specifically:
1. You claim all rules are applicable to empty sets and to non-empty sets.
2. You use one specific way of showing how on specific the rule applies to non-empty sets.
3. You use another, different specific way of showing how a different specific rule applies to empty sets.
4. You claim that the rules you used in 2. and in 3. are not only compatible, but point at the same result.
No, they don't.
This was illuminated first in your Venn diagram example, where inside the circles the meaning was only meanigful if something existed, while outside the circles it was meaningful only when nothing existed (in the set).
This was illuminated in your second example, when you used a whole bunch of negations to arrive at your points, but each of the three sets used different negations of different things. It would have only been meaningful if you used the same logical steps in all three scenarios and arrived at the same conclusion, that is, at a unified rule. But you did not.
To be completely honest, I did not read your third explanation yet, I'll do it later. But I don't know why you don't see that your methodology does not cut the mustard, so to speak. If the same rule only applies to empty sets when one condition is met but the other condition is not met, and the same rule only applies to non-empty sets when another condition is met, but not the first one, then it's not the same rule, but a modification of the same rule in the two separate instances. And if you modify a rule so it becomes different from its original form, then it is not the same rule.