What is the difference between cooking pottery: ceramic, porcelain, stoneware, flameware, earthenware/terracotta, & pure-clay?

Firing temperatures.

None of the clays, when properly fired, have any influence on the health of the food. Different ones have different heating requirements if used for cooking food as some will crack if shocked from cold or room temp too quickly to oven temperature. "pure-clay" is a nonsense term that does not connect with others which may be purely one chemical composition of a natural clay or a blend for better performance. All fired clays fall within the category of ceramics. Look up the simple definition of the middle 4 types on Wikipedia and detect what you think is important

Google it locally obviously