Complex sentence: Either convention will work, since the primary significance of the Minkowski inner product is that for any two four-vectors, its value is invariant for all observers; a change of coordinates does not result in a change in value of the inner product.
Split into simpler sentences: Either convention will work, since the primary significance of the Minkowski inner product is that for any two four-vectors, its value is invariant for all observers. An important property of the inner product is that it is invariant, i.e. a scalar: a change of coordinates does not result in a change in value of the inner product.