Agree with you OP, I cannot stop referring to it as clay, hardened clay or stained clay, it just rolls of the tongue in a way that terracotta refuses to! I recently played with someone, a new player, who had only ever known it as terracotta and they couldn't figure out what I was going on about, what is this 'hardened' clay you keep talking about..? lol
Well terracotta is still a type of clay. But as far as i'm concerned, you make things with terracotta, you *play* with clay. As an added bonus I found its a lot easier to search terra cotta than it is to search clay.
I have to use both hands to type clay. I type "terr" with my left hand and there it is. Hope that helps.
I might suggest that clinging to the idea of certainty is not a very good idea since things tend to change a lot.
Honestly, you could probably just keep calling it "stained clay" and everyone will still know what you're talking about. Kinda like how I could say "adminium," "sulfur," and "reeds," but people will still know what blocks & items I'm referring to.