I think this is really a question about the terminology. The concept of "the elastic component of the residual strain" may be confusing, until you take it apart to see what it really means.
The total strain in the material is well defined - it is what a strain gauge would measure.
If the structure has undergone plastic deformation, the total strain may be non-zero even when there are no applied loads - in other words, the structure has been permanently deformed from its initial shape. That is what residual strain means.
For continuum mechanics, it is useful to split up the total strain into different components, such as plastic strain, creep strain, thermal strain, elastic strain, etc. These are theoretical concepts, and by definition the elastic strain is the only component which is related to the elastic stress.
As an example that doesn't involve any nonlinear or time-dependent behaviour, consider a bar of material with both ends rigidly fixed, where the bar is then heated. The total strain is zero, because the length of the bar remains constant. However there is a non-zero thermal strain (equal to what the thermal expansion would be if the bar was not fixed) and an equal-and-opposite elastic strain which makes the total strain zero. The stress in the bar depends only on the elastic strain.
So to summarize the answer, "the elastic component of the residual strain" and "the residual elastic strain" are two names for exactly the same thing.