Curvature is not "measured with respect to [an embedding] space". If you have a triangle and the sum of the angles is not 180 degrees that's a non-Euclidian (curved) space.
But that's not how the word "curved" is used in everyday life. Call it non-Euclidian if it's non-Euclidian. Redefining common words only leads to confusion.
Participating in a discussion about an article titled "the quantum source of space-time" to say that basic concepts are nonsense because some words do not have the same meaning as in everyday life also leads to confusion.
Take a 50-kg object (110 pounds) and carry it in your arms for an hour on a flat road. You will have done no work against gravity. You may complain that this is not how the word "work" is used in everyday life - and that you will have done a lot of "work" against gravity (preventing the object from falling to the ground). However, if you wish to discuss Physics and Mathematics, using terms that have precise definitions in those fields, you cannot object on the basis that these words do not have the same meaning as non-Physicist and non-Mathematicians ascribe to them.
Why stop there? Field, black hole, wormhole, string, force, spin, colour, charm, strangeness. All of these terms and more are used by physicists and have very different meanings to those used in "everyday life".