I've heard that positivism aimed to be purely empirical, while logical positivism recognised that empiricism needed to be combined with logic in order to actually be able to figure out anything. Is this an accurate characterisation of these two movement?
No, it's not.
The basic ideal of positivism, both old and new, was to purify science by eliminating from it all speculation (the positivists identified all speculation with "metaphysics", in a derogatory sense). The older kind of positivism was just more humble. It was a philosophy of science. A typical project of the old positivism was Ernst Mach's attempt to rewrite Newton's physics without assuming absolute space. With logical positivism, the stakes got raised. It was emboldened by new developments in the foundations of logic, and mathematics, which seemed to promise new and scientific foundations for language generally. And with verifiability as a criterion of meaning, not just of good science, logical positivism posited itself as a basic philosophy, not just as a philosophy of science. The logical positivists were indeed occupied with logic, that is inductive logic, the logic of discovery, empirical verification and probability.