It sounds to me like you're unaware of (or dismissive toward) the existing discourse around the topic of neoliberalism. Mises.org may view the term as a meaningless epithet leveled against their ideological kin by ignorant or irrelevant onlookers, but one can't simply hand-wave away the existing literature on the topic. Redefining existing terms to suit one's own goals, perhaps through the use of arbitrary purity tests, is unhelpful at best. As you acknowledged, "the incentives for distortion are clear."

Moving on, I'll reiterate that Hayek is widely considered a (very significant) neoliberal. This isn't a controversial viewpoint - check out neoliberalism's entry in Wikipedia (https://en.wikipedia.org/wiki/Neoliberalism) or the Stanford Encyclopedia of Philosophy (https://plato.stanford.edu/entries/neoliberalism/) for the mainstream perspective on his role in its development. The notion that Hayek is not a neoliberal, however, would seem quite controversial to most in this space.

> In practice neoliberalism does not represent the free-market as presented by Hayek.

Keep in mind that Hayek was a theoretician, so his ideas represent a starting point, not a conclusion, when it comes to enacted policy. What are some good examples of neoliberalism in practice that you find wholly incompatible with Hayek's views?

> Take the example of neoliberal negotiated "free trade" agreements and contrast that to laissez-faire unilateral free trade.

While I can appreciate the differences between unilateral free trade and "neoliberal negotiated 'free trade' agreements," I'm not sure how the existence of those differences removes Hayek's prominent role in the history of neoliberalism. I'll also point out that the (confusing) catch-all use of the word "neoliberal" to describe American center-left politics from Carter onward shouldn't be conflated with academic use of the term.

> Mises was also a founding member of the MPS, will you have me believe that he was a neoliberal?

While I probably wouldn't primarily characterize Mises as a neoliberal, he had a huge influence on Hayek and maintained many views that are compatible with neoliberalism.

> Neoliberalism is distinct from liberalism insofar as it does not advocate laissez-faire economic policy but instead is highly constructivist and advocates a strong state to bring about market-like reforms in every aspect of society.

This is from economic historian/philosopher Philip Mirowski, I believe, and should be viewed in that context. He's describing an existing neoliberal political and economic system in toto, as opposed to the theoretical principles that Hayek or Friedman would have discussed. While one might discuss policies intended to increase economic freedom as if they could be overseen by a small, limited government, and would result in universally-increased freedom, actually implementing them could require a powerful and oppressive dictatorship like Augusto Pinochet's in Chile.