You will have more luck getting a roomful of Biologists agreeing on a definition for the word ‘Alive’, or Logicians for the word ‘Reason’, than you will with a roomful of Psychotherapists defining the word ‘Self’.
The Bio-Engineer knows it is a silicon-chip atop the neurons and between the firing synapse. The Geneticist sneers at this simplification at what is clearly a Gene [imminent in its discovery].
We won’t even broach the Mystics for now. But a particularly famous one from India is: ‘The sense of ‘I-ness”, which means whatever you want it to mean.
We go low-tech. We ask the Grammarian. ‘Self’ is a complicated idea; so we start with ‘Subject’.
‘The Subject of a sentence is the person, place, thing, or idea that is doing or being something. It is what acts or is acted upon.’
Ego is Latin for ‘I’. The Cambridge definition reads: ‘Your idea or opinion of yourself’. Note the loop.
This ‘I’ business is covered in numerous later Posts highlighting its historical evolution from Fiction to Philosophy. But I present here an excerpt from my copy of the translation of the Vishnu Purāṇa  by the remarkable Horace Hayman Wilson [First translator of the Rig Veda and the first occupant of the Boden chair for Sanskrit at Oxford]:
‘The sense of Ahankára cannot be very well rendered by any European term. It means the principle of individual existence, that which appropriates perceptions, and on which depend the notions, I think, I feel, I am. It might be expressed by the proposition of Descartes reversed; ‘Sum, ergo cogito, sentio,”