Why do convex and concave function have their own name like that not the opposite?
I don't know the historical origins of the mathematical usage, but I have always thought it was backward also, and I'm a native speaker of English. The terms in mathematics are relative not to the x-axis, but to a horizontal line above the curve (at $y=\infty$ if you like).
To avoid confusion, I think it's often better to use the terms "concave up" and "concave down", which are quite clear ("convex" = "concave up", and "concave" = "concave down").