There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean.
Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense.
When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless, if the sum of the an exists, it is equal to the limit over r. This therefore fits exactly into the abstract picture.
The original result of Alfred Tauber (1897)[2] stated that if we assume also (see Little o notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent.
In the abstract setting, therefore, an Abelian theorem states that the domain of L contains the convergent sequences, and its values there are equal to those of the Lim functional.