I'll jump in here. I think, as usual, there is a sort of vocabulary issue in that different writers from different disciplines use similar words in slightly similar ways.
In most psychological writing, there is little discussion of spirituality. (Not religious, just spiritual in the sense of things beyond traditional physical measurement (for lack of a better word)). I think perhaps transpersonal psychology and Internal Family Systems go beyond the concept of "self" as a cognitive construct, whereas most psych writing sees it as purely cognitive--as in the "self" is the larger set of neural networks that make up the whole personality. The sum of the parts as it were. The brain as a kind of "container" of our personality states. All people are naturally multiple in terms of having a variety of parts of their personality. However, some people's parts seem to work together a bit better than others'.
When we're talking about dissociative states, the term "self" becomes wildly problematic and confusing. Because many of us feel we have more than one self--even those without one of the complex dissociative disorders. So, for instance, I have parts that fully believe they are my self. When I am in one of those parts, it feels like me, myself. But it is not. It is a part of myself. Not my whole self. This is a really important distinction. These parts can be what van der Hart et. al. call ENPs and ANPs, and others call other things.
In my rather tangled understanding, the term SELF is an important identity construct, whether we understand it in a cognitive sense, a spiritual sense, or some combination of both. Cognitively, the SELF would be that state that monitors the inner and outer world and responds flexibly. Functioning somewhat like executive function...rallying all the "players" to work together to achieve a goal. When that SELF is compromised, parts jump in to wrestle for executive control.
The term HIGHER SELF or LARGER SELF is related, but a bit differently, in that it acknowledges an identity that is located in, but also exists beyond my brain/body. I think I read somewhere that it is a sort of metaphysical identity.
I actually like to think of difference between the self and the larger self as a difference in the focus of relational activity. The SELF of most psychology is limited to what is happening inside the brain. It acknowledges that the SELF can be affected by and can affect others, but it is fundamentally--well--selfish. LOL. It's kind of our traditional understanding of "self." The OED defines self as
"A person’s essential being that distinguishes them from others, especially considered as the object of introspection or reflexive action:"
The LARGER SELF exists both within and beyond the brain. It is the self PLUS. Not only is it the traditional self, it is also connected to others and energies in the universe in ways that scientists are just beginning to acknowledge and understand. And it is in not only the brain, but the whole body (if you read anything about chakras, and meridians, and the yogic layers of the self this will make more sense).
There seems to be some sort of western cultural shift of understanding about personhood that has been happening in the past 20-30 years. Perhaps it is the influence of Eastern understandings that began to gain more widespread traction in the 1960s. I'm no anthropologist, so I don't really know. In my understanding, the idea of SELF and HIGHER SELF are attempts to articulate a state of being that exists both within and beyond the brain and body of an individual. A kind of meta-self if you will, that connects us with everything outside of us in space and time. This concept can feel a bit much for some people because it is hard to grasp and gets into metaphysics.
It is so important to psychological healing, however, that it is worth delving into an understanding of it. Because, IMHO, it is what offers hope for the loneliness and alienation and wounding that we all experience. The possibility of healing connection with other people, with the natural world, and with the "divine" (in the broadest and most inclusive sense). When we can tune into the larger self, we tune into compassion and creativity and connection, etc.. Tuning into the larger self is (at least for me) what brings joy amidst all the suffering and turmoil of my existence.
Richard Schwartz, founder of Internal Family Systems therapy, calls SELF the "larger self" and describes it here:
http://www.selfleadership.org/the-larger-self.html
Wikipedia has a decent if simplistic overview of the concept of "higher self":
https://en.wikipedia.org/wiki/Higher_self