I'm afraid you're right: there's no simple answer.
We know that in the British navy, at least, it was by the middle of the eighteenth century a matter of regulation that all line supplied to the navy be soaked in Stockholm tar. I've never seen Stockholm tar, but I have it on good authority that it was a warm, medium brown in color. Standing rigging, by that time, apparently was routinely coated, after it was set up, with a concoction containing tar, lampblack, and various other chemicals; if it wasn't pure black it must have been close.
The old "Board Room" models of English warships prior to the mid-eighteenth century generally have rigging that's all one color. It's tough to figure out what the color originally looked like, since it's darkened so much with age, but rarely if ever does one find two colors of rigging line on such a model. I once discussed that with David Lyon, who for many years was in charge of the plans collection at the National Maritime Museum in Greenwich. He flatly denied that there was ever any distinction between the colors of standing and running rigging in the Royal Navy; he dismissed that distinction as "some sort of American model builder's convention." Frankly I think he was mistaken, certainly as far as the period after about 1760 is concerned.
Prior to that time I honestly don't know. The use of tar, with various other stuff mixed in, for preservation of various materials against the weather seems to be an extremely old idea. (Recent research on the seventeenth-century Swedish warship Wasa seems to have established that her entire hull, from the keel to the upper wales, was coated with tar, and tar as a coating for underwater planking seems to be older than that.) Donald McNarry, whose ship models I personally revere above all others, makes all his rigging out of wire, and paints it; according to one of his articles, he uses Humbrol RAF dark earth (actually a dull, medium brown) for running rigging and Humbrol "track color" (a rusty black made for model railroaders) for standing rigging. He knows about as much about such topics as anybody I know of. My inclination is to make the color distinction between standing and running rigging in ships from about the beginning of the sixteenth century onward, but I'm by no means certain that's right.
I've thought for a long time that a thorough study of the color of rigging line, in various nations throughout the ages, would be a worthwhile thing. Unfortunately, though, I'm not sure there's actually enough evidence to do it with any real confidence.