But we shouldn't define One True Approximation, which is oxymoron, there is no true approximations.

What we need is to define practical useful approximation. For example I want to create loose copy of me during uploading, and some information will be lost. How much and what kind of information I could skip during uploading? It is important practical question.

I would also add that there is two types of identity, and above I spoke about second type.

First one is identity of consciousness (it happens if you lose all your memories , overnight, but from our last discussions I remember that you deny existence of consciousness in some way).

Second is identity of memory where someone gets all your memories, thus becoming your copy. (There is also social, biological, legal and several other types identities).

They are not different definitions of one type of identity, they are different types of identity. During uploading we have different problems with different types of identity.

Me-tomorrow will be similar to me-now, and it is governed by second type of identity, and there are no much problems here. The problems appear than different types of identity become not alined. In Parfit's example if I lose all my memories and someone gets all my memories, where will be I?

(Parfit named this two types of identity "numerical" and "qualitative", the names are not self evident, unfortunately, and the definition is not exactly the same, see here http://www.iep.utm.edu/person-i/:

"Let us distinguish between numerical identity and qualitative identity (exact similarity): X and Y are numerically identical iff X and Y are one thing rather than two, while X and Y are qualitatively identical iff, for the set of non-relational properties F1...Fn of X, Y only possesses F1...Fn. (A property may be called "non-relational" if its being borne by a substance is independent of the relations in which property or substance stand to other properties or substances.) Personal identity is an instance of the relation of numerical identity;")