Its true that many of the words used are unregulated. Generally speaking the more specific the better. For example, ‘cage free’ is very vague - essentially meaningless. But “500 per hectare” is more specific.

Without making yourself an expert in this, it’s pretty hard to know what is ‘good’. So in the end, it’s probably easier to find a reliable trustworthy supplier whose eggs are high quality, and stick with them. (And it probably isn’t going to be from one of the duopoly supermarkets.)