"Self" is a cognitive error. Mahayanasutralamkara says:
In itself, the view about a self lacks a characteristic of a self,
As do its deformities-their characteristics differ [from a self].
Nor is there another [self] apart from these two, so it arises as a mere error.
The "deformities" are the skandhas, an examination of which will reveal that their nature(s) is/are contradictory to the notion of "self"--and even the collection, when considered as a whole, is contradictory to what we "feel" a self to be.
The "view about a self" is a sort of conceptual construct about a "self"--it is a mental event, as you say. This is the "self" that we can talk about as "soul," or "essence," or whatnot, and which has no existence whatsoever, and is completely imaginary. But there's another, more subtle, kind of mental 'event,' or "mental element," if you will, which is a sort of seemingly-fundamental and almost pre-consciouss "self referential" attitude underlying all discursive thought.
It's that more subtle element which is almost an assumption, not something we really analyze or think about conceptually, that is the real error--not some sort of philosophical construct, but a pyschological 'mode' of sorts. This is the "Afflicted Mind," spoken of sometimes as the 7th consciousness--with the 5 sense consciousnesses, and the 6th, or mental consciousness.