Not exactly. Supermarkets also jack up prices without any improvement at all.
I.e. better conditions require higher prices, but higher prices can mean better conditions or more supermarket profit. And the supermarket is incentivized to pick profit, together with pretty pictures and words that "suggest" better conditions.
Which is why I don't generally trust the wording on packages with regard to animal conditions. I'm not an expert in which exact phrases legally mean substantially better conditions, vs. which ones sound good but aren't meaningful at all. Nor should I be expected to.
I'd much prefer the government just legislated conditions that are humane. Either animal welfare matters or it doesn't. It doesn't make any sense for it to depend on individual consumers. A few people buying top-tier eggs isn't ever going to improve anything for the vast majority of hens.