Under a free market one voluntarily buys protection, negotiates with relevant parties how arbitration is to be conducted, and so on. These are all part of the market just the same. A "free market" that isn't free continues, despite your insistence, to be nonsensical.

When was the last time you made an offer to your police force? Under your free market definition it is necessary for you to express your demand position, else supply and demand cannot be determined. Yet, I suspect your answer is never. There isn't a free market for you to express that in. It is not something you have to think about as the single desk has already sorted it for you.

What you are thinking of, again, is a mixed economy, where the government has taken control and regulates some submarkets. That is how our world functions. A market having unconstrained submarkets that see prices determined by supply and demand isn't the same as a free market. What to call the submarkets within a mixed economy that are based on supply and demand isn't generally accepted as there isn't usually much reason to need to talk about it, but I posit "unregulated market" remains the leading contender for what shows up most often when one is trying to single it out. "free market" is a non-starter as it already means something else in the same context, as seen in the very definition you gave us.

You are, of course, revealing one of the reasons why we've never seen a free market. The concept is full of unresolved problems that make the whole thing untenable in practice. I think we can agree it is almost certain that people wouldn't be able to handle themselves if there wasn't a government automatically providing those things for them. But that's the nature of a thought experiment—It doesn't have to be realistic.