The Von-Neumann Morgenstern axioms talk just about preference over lotteries, which are simply probability distributions over outcomes. That is you have an unstructured set O of outcomes, and you have a total preordering over Dist(O) the set of probability distributions over O. They do not talk about a utility function. This is quite elegant, because to make decisions you must have preferences over distributions over outcomes, but you don't need to assume that O has a certain structure, e.g. that of the reals.

The expected utility theorem says that preferences which satisfy the first four axioms are exactly those which can be represented by:

A <= B iff E[U;A] <= E[U;B]

for some utility function U: O -> R, where

E[U;A] = \sum{o} A(o) U(o)

However, U is only defined up to positive affine transformation i.e. aU+b will work equally well for any a>0. In particular, you can amplify the standard deviation as much as you like by redefining U.

Your axioms require you to pick a particular representation of U for them to make sense. How do you choose this U? Even with a mechanism for choosing U, e.g. assume bounded nontrivial preferences and pick the unique U such that \sup{x} U(x) = 1 and \inf{x} U(x) = 0, this is still less elegant than talking directly about lotteries.

Can you redefine your axioms to talk only about lotteries over outcomes?

You started out by assuming a preference relation on lotteries with various properties. The completeness, transitivity, and continuity axioms talk about this preference relation. Your "standard deviation bound" axiom, however, talks about a utility function. What utility function?