We do not have to extract the rules from the view examples we have. Tolkien reported the decision about the half-elves very accurately [
The history of Middle-earth; volume V:
The Lost Road and Other Writings: Language and Legend before The Lord of the Rings; part two:
Valinor and Middle-Earth before The Lord of the Rings; chapter VI:
Quenta Silmarillion; sub-chapter:
The conclusion of the Quenta Silmarillion; paragraph 9]:
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Then Manwe gave judgement and he said: 'To Earendel I remit the ban, and the peril that he took upon himself out of love for the Two Kindreds shall not fall on him; neither shall it fall upon Elwing who entered into peril for love of Earendel: save only in this: they shall not ever walk again among Elves or Men in the Outer Lands. Now all those who have the blood of mortal Men, in whatever part, great or small, are mortal, unless other doom be granted to them; but in this matter the power of doom is given to me. This is my decree: to Earendel and to Elwing and to their sons shall be given leave each to choose freely under which kindred they shall be judged.'
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The children of Elrond would have had no choice if it would not have been granted to them by Manwe. It is an interesting question how they and their family did get that information, but it is reported by Elrond to Aragorn in the case of Arwen, so they did know.
Anyhow this does not say anything about Dior. But I agree with Puddleglum. Dior was the son of Beren and Luthien while both were mortals, so he was a mortal. That he might not have been happy with that is clear, but to marry an elven woman was his choice as is it was the choice of Imrazor who married Mithrelas. Neither of these couples could change the doom of separation as long as Ea my last or even longer (practically for ever).
About Gandalf confusing the 3 with the other great Rings in his explanation to Frodo: Is it at this point necessary for Gandalf to be absolutely precise about this feature? I don't think so. He is to explain Frodo which of the great rings the one Fordo has is. The 3 are accounted for, but he is not permitted to speak about them. And Gandalf and Frod do know that Frodos ring did make the wearer (at least a mortal) invisible. The important point is to warn Frodo about the 'side'-effect of using the ring often (fading to permanent invisibility). It would be completely useless to tell Frodo that he would not be in that danger would his ring be one of the 3.
Since the 3 are accounted for, the effect that the ring did confer invisibility to Fordo and Bilbo is an evidence that it is one of the great and that is a farther point why Gandalf does mention it here. After giving farther evidence Gandalf doe then make the final test and proves the identity of Fordos ring. So it is at this point (and also later at the council of Elrond) unnecessary to explicit explain the whereabouts and exact properties of all the other 18 great rings. (BTW this is a petty, as I would have like a full account of where and when the 4 dwarven rings that Sauron had not collected were destroyed and how the Wise did know.)
Back to the original question: Would Half-Elves become invisible wearing one of the Rings that conferred invisibility?
I think it depends. Elrond: No. When he had a chance to lay hand on one of these rings he had already chosen to be an elf. So by all means he was.
Arwen: I think her choice was made effective when she married Aragorn. Otherwise Elrond delaying of the marriage would be useless. So I would say up to that point she had the life and vigor of the Elves and therefore would not become invisible. (If Elrond would have allowed her marriage earlier, I would think she would have become invisible wearing one of the rings in question, but that chance never came.)
Elladan and Elrohir: Since we know very little about the doom spoken about them we can only guess by the similarity with Arwen. So I think they wouldn't become invisible, since we are told in one of the letters that they probably delayed their choice even after Elrond had left Middle-Earth. So as long as the rings were functional they still had the life and vigor of the elves.
Earendil and Elwing: They did not have any chance to get hold of one of the rings at all. (But if there would have been a chance, I think they would not become invisible.)
(If we ignore for the sake of speculation the time line: Elwing, Earendil, Elrond and Elros before their choice: Difficult, but I would again go with the similarity of Arwen => as long as the doom is unclear the life and vigor of the elves is granted and so no invisibility.
Elros after his choice: See again the similarity with Arwen. Once he was a mortal he would become invisible wearing one of the rings in question.)
Respectfuly
Findegil