Great question. If you'll permit, I'll turn your question on its head. Instead of asking about utility functions (that are ultimately subjective), I'll answer looking at implicit assumptions about optimal strategies, making either assumption. I hope you can see that ultimately and intuitively this ends up in the same place; but just approach the problem from the opposite direction ;-)
Put simply - both are "industry standards", for different subsections of the "industry".
Let us assume the cliche of a two-asset world, with all the cliched assumptions about normal return distributions etc. Let us, 100% for convenience's sake, then just call these assets "stocks" and "bonds".
If I position proportional to inverse volatility, this is called "risk parity". It's a bona fides strategy. Hundreds of billions do it explicitly. You're getting into the trillions those who do it implicitly. It's a thing. If you need, I can find you some Fed minutes talking about this being an issue worthy of policy discussion; because I was called in (from UK) myself to discuss this in a purdah period. Google "Bridgewater All Weather" if you want to go further down the rabbit hole...
If you take this blue-pill as an investment strategy, you end up making a very simple investment assumption. You are assuming that the excess return of all assets in the portfolio is proportion to volatility, ie that all assets in the portfolio have the same Sharpe Ratio.
[For large numbers of securities, one can obviously discriminate between inverse vol and inverse marginal contribution to risk; but the concept is hopefully clear enough]
If I position-size relative to variance, then I am making a very different bet. As luck would have it, the optimal (from a return-maximisation perspective) position size for any normally-distributed set of returns is... mu/variance. It's the "Kelly Bet" for a continuous normal asset.
So inverse-variance positioning would be testament to betting with high conviction that stock returns >0; but having no pretense of any idea why some stocks might be better or worse than others. Relative to the index, my bets would carry zero assumptions about any single stock; or its relationship to any other stocks in the index.
Practically, there is no evidence that high-vol stocks outperform low-vol over the long-run. The evidence suggests maybe the opposite (in defiance of the CAPM). If so, best run your research against vol rather than variance. The latter only works for multi-asset people who are 100% comfortable using futures markets for shorting/leverage...
hope this helps.