The wikipedia definition isn't the clearest definition of logical positivism. If I had to define logical positivism in one sentence (which, in philosophy, is almost universally a bad idea), I would say something like this:

Logical positivism is a philosophical movement which maintains that statements are only meaningful if they can be formally derived or empirically verified.

In that respect, I think that logical positivism and the kind of empiricist approach identified by the OP have a significant amount in common. Similarly, I think the approaches share many of the same problems.