Here's what I know for sure - When the mini-series was repeated in the UK in 1980, it had the split-screen titles on at least three of the episodes - these were eps 1,3 and 5. I remember that the first things I noticed were the smaller logo and the different arrangement of the music to what I remembered from season 1. I can't say about ep.2 as I missed the start, and the BBC never showed ep.4. I would think that the BBC were using the same prints as they had for their first run in September 1978 and I think I vaguely remember the split-screen titles right from the start, so assuming my memory's correct, and that the BBC didn't get new prints in 1980, that means the split-screen titles were put on in 1978.
As for the titles posted here, I don't think that this is the complete sequence. What we have here was taken from an edition of Wogan, shown on Christmas Eve 1986, with Larry and Linda as guests. They wouldn't have shown the whole of the titles in a chat show. It was followed by a clip from the first episode, where the family have gathered and JR insists on going outside with Pam. You can hear the start of it at the end of the clip here. I have to disagree with Chris2 that it's the closing theme we here. It's not, it's actually the opening theme, which again makes me think that this is only part of the titles, not all of it.
I think that two complete titles were made for the mini-series to give the producers a choice of style, including the logo - the one that we have a part of here, with the familiar logo but in white/faded yellow, and the split-screen/small yellow logo. Here I agree with Chris2 in that it is unlikely that the split-screen titles were made later with its different logo to the one that we would already have been familiar with. I also think that originally the one here was put on the first three episodes, as all the other credits are white/faded yellow on those episodes (the writer and director credits over the first scene, the freeze-frame credit and the end credits). This is also in keeping with the fact that the background to the end credits also changes (still shots for eps 1 to 3, the moving aerial shot of the ranch for 4 and 5).
Whether the first three episodes were actually ever shown with these credits or they were changed before broadcast to the split-screen, I can't say.