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.
I can see where it got that name actually, I guess the bedrock layer was seen as the "admin" layer on servers seeing as only admins could break it with plugins.
And yes, I still call gunpowder "sulfur", and I'm actually considering renaming it to that in my resource pack that I'm going to make for 1.13.
Bedrock was also only able to be placed and broken by admins back in Classis, when it otherwise didn't really exist as a block, as the bottom layer of a world wasn't really made up of blocks, but only used the Bedrock texture IIRC (This means that it wasn't possible to place blocks on the bottom layer).