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Vaniver comments on Terminal Bias - Less Wrong
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Rational agents should be. Irrational agents - in this case, prone to risk aversion - would instead be willing to pay a finite cost for the bet to be replaced with the sure deal, thus losing utility. You can fix this by explicitly incorporating risk in the utility function, making the agent rational and not risk-averse any more.
Again, what you are saying is a non-standard definition. The commonly used term for the bias you're describing is certainty effect, and risk aversion is used to refer to concave utility functions.
First, concave utility function is just a model for risk aversion which is "the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff." (wiki)
Second, the certainty effect is indeed one of the effects that is captured by my preferred model, but of course it's not limited to it, because it's possible to behave in a risk-averse manner even if none of the offered bets are certain.