There are many different views of Self. Looking at DN9 Potthapada Sutta already give you a sample of some. If you ask various people now, they would probably postulate a Self differently each, even if they come from the same religion.
What matters more is to understand what Self meant to the five ascetics, how the Buddha refuted them so that we may utilise them in the same manner when we have the same views.
As a basis for existence, the five aggregates can be seen as belonging to Self or are Self.
If one view that they belongs to Self, Buddha refuted it categorically as they cannot be controlled.
First Argument
Mendicants, form is not-self. For if form were self, it wouldn’t lead to affliction. And you could compel form: ‘May my form be like this! May it not be like that!’ But because form is not-self, it leads to affliction. And you can’t compel form: ‘May my form be like this! May it not be like that!’
If they are existence itself, the Buddha asked the ascetics to consider the fact that they are impermanent. It follows naturally that they cannot be satisfactory. Just think of this body and mind that grows old, get sick and dies. Can you develop a sense of security and satisfaction in it? Is it a good candidate befitting for Self? If not, what would be the most skillful mean to go about considering it?
Second Argument
What do you think, mendicants? Is form permanent or impermanent?”
“Impermanent, sir.”
“But if it’s impermanent, is it suffering or happiness?”
“Suffering, sir.”
“But if it’s impermanent, suffering, and perishable, is it fit to be regarded thus: ‘This is mine, I am this, this is my self’?”
“No, sir.”
“So you should truly see any kind of form at all—past, future, or present; internal or external; coarse or fine; inferior or superior; far or near: all form—with right understanding: ‘This is not mine, I am not this, this is not my self.’
In fact the Buddha presented a third way to view the relationship between existence and the five aggregates. When there is craving for the five clinging aggregates, it becomes the fuel for existence. With the cessation of craving, the fuel ceased to be and extinguishment follows as a result.