I used to call these sorts Trinity theories of “modalism.” I still think that is not a bad description, so long as you specify that it is not by definition heretical or wrong.
But I now think “one-self” is a more neutral term, which even better describes the sort of theory.
The reasoning goes like this.
The Christian God must, after all is said and done, be a god – a mighty self. He has plans, knowledge, and loves us. And in the Bible, he almost always says, “I”, “me” and “mine.” Further, there is there is only one God. And the Christian God just is the Trinity. Therefore, the “persons” of the Trinity must not be real persons, that is, selves. There’s only one of those – it is the Trinity – the one God himself.
That is how you get to a one-self Trinity theory – you demote the persons of the Trinity – either all three of them, or the Son and Spirit, to less than selves. That is, you make them in some sense modes of the one God (or of the Father). The whole who point is to end up with just one divine self, one god, God.
You see this reasoning, for example, in certain Reformed sources, like the ones Mr. Phil Gons is reading – here, and here. But you see it across the catholic spectrum, in my view. This is why theologians often say that many Trinity theories “tilt towards” modalism. Well, many of them just are modalisms – that is, one-self theories. You can announce that the relation between the one divine being and the “persons” is mysterious till you’re blue in the face, but if after all that, you conceive of God as a self, you have a one-self view – NOT something which is tilting, sliding, leaning, sneaking, fox-trotting, or tending towards a one-self theory.
Is it orthodox? It can be – just specify that the modes are all eternally concurrent, and that they are essential to God. Mainstream theologians are generally agreed that this is sufficient to make it a non-heretical modalism, to make it, in standard lingo, not “Sabellian.”
But there’s a huge problem with all of this; the Son of God is clearly a self too (specifically, a human self, a man). And he’s got to be a different self than God, because he’s the one mediator between God and humankind. To me, this is enough to show that the theory is inconsistent with central and clear claims of the Bible.
This also, I think, provides some of the impetus towards three-self Trinity theories, aka “social” theories.