On page 12 in R.J. Williams' Introduction to the Mathematics of Finance, the author states that under the risk neutral measure, the discounted underlying stock price is a martingale.
He then follows with an explanation why this measure is called "risk neutral", which I cannot follow. More specifically, he uses an analogy to risk neutral persons, but I do not understand what the risk neutrality of the described "risk neutral person" has to do with the properties of the so called "risk neutral measure":
A person is said to be "risk averse" if the person prefers the expected value of a payoff to the random payoff itself. A person is said to be "risk preferring" if the person prefers the random payoff to the expected value of the payoff. A person is "risk neutral" if the person is neither risk averse nor risk preferring; in other words the person is indifferent, having no preference for the expected payoff versos the random payoff. The probability $p*$ [the setup is a discrete time single phase binomial model, where $p$ is the true up-probability of the underlying] is called a risk neutral probability because under $p^*$, a risk neutral investor would be indifferent to the choice at time zero of investing in stock or bond, since both investments have the same expected payoff at time $1$ under $p^*$.
In my understanding, stock and bond having the same expected payoff is a prerequisite for the differences of the three described persona to even become appreciable, so I don't understand why the assumption that this prerequisite is satisfied would be connoted to a single one of these persona. In other words, the definition of "risk neutrality" seems to be a matter of expectations (since it entails a martingale property), whereas risk aversity or risk preference as described in the quote are a matter of variances and I don't see their connection. You can still be risk averse or risk preferring, even when the discounted stock is a martingale (i.e. you have a risk neutral measure).
Can anyone help me get a better understanding of the quote and/or the term "risk neutral"?