[William Cloud Hicklin]

We can't base anything on the Keepers of the Three, because the Three were different. "They did not confer invisibility." Thus spake Tolkien.

The OP brought up 'unsheathed swords.' This apparently is a reference to The Hobbit, where Sting's 'light' is visible as the Ring-wearing Bilbo attacks the Spiders. One could of course just dismiss this as being The Hobbit, when Bilbo's ring wasn't yet The Ring. But we might combine it with another quasi-canonical text, The Disaster of the Gladden Fields, and conclude that if a person bearing a (magical?) light-source puts on the Ring, he and everything he wears or carries becomes invisible- but the light escaping from the source becomes visible as soon as it leaves. This would explain why the Elendilmir continued to shine until Isildur covered it with his hood, and why Sting shone when drawn but not when sheathed in the presence of enemies.

It does seem that a Ring affects that which its wearer is wearing or holding at the moment he puts it on, but not items picked up subsequently. The Nine's original mail and helms etc became invisible as a product of their ring-induced fading; but the cloaks they put on subsequently, and the physical weapons of the Witch-king, remained visible. Of course this could just as well be explained by the fact that by the end of the Third Age they had long ago handed over their Rings to Sauron.