I think you are confusing the mid-20th-century (1930s-1960s) use of "neoliberal" with the late-20th-century-to-current (1980s-and-later) use. The term basically fell out of use and was resurrected with a completely different meaning.

It was associated more with Pinochet in Chile and the US policy preference which favored Pinochet-like figures.

(Clinton in the 1990s saw the adoption of the new term "Third Way" for something which was somewhat similar to the mid-20th-century use of "neoliberal" -- and quite distinct from the then-recent resurrection of the term -- though the contemporary left-wing social and right-wing economic positions it was synthesizing -- particularly the left-wing social positions -- were somewhat different than those of mid-20th Century neoliberals; though there is a connection to the 1980s definition of neoliberalism, since the then-current right-wing economic position that was key in the 1990s Third Way was the new, 1980s-definition, neoliberalism. But 1990s [and later] Third Way-ism isn't the defining instance of 1980s-and-later neoliberalism, its just the most politically left-wing movement incorporating some elements of 1980s-and-later neoliberalism.)