They must have been using different crates for you (different region or perhaps era). For me they were standard plastic bins[1] with a separate “cold bag” inside for frozen stuff. No actual styrofoam I recall, although this was also over 10 years ago so I could be misremembering.
The style of plastic bin definitely looks the same. The ones we were getting looked something like this[0], same folding-flap top as in your link but form-fitted insulation inside:
Yeah. I’m not certain if the Amazon ones were actually the same as the ones I linked. But extremely similar at least.
It’s been a long time. Very plausible that we did get the ones with the styrofoam sometimes and I just don’t remember. I know we got the cooler bag sometime.