So how does this extend the concept of a base? At the moment it doesn't - it just looks at it from a different angle.
But what about other similar reactions of ammonia or water, for example? On the Lewis theory, any reaction in which the ammonia or water used their lone pairs of electrons to form a co-ordinate bond would be counted as them acting as a base.
Here is a reaction which you will find talked about on the page dealing with co-ordinate bonding. Ammonia reacts with BF3 by using its lone pair to form a co-ordinate bond with the empty orbital on the boron.