x=x(t)

y=y(t)

Is not a space curve. It is a "space curve," ie,calleda space curve. "Distance," "Tangent," "Normal," (& "BiNormal" in 3d) and "curvature" are defined mathematical operations which have their origin in and can potentially be interpreted as real quantities. A space curve can be created by assigning 1 to a distance on a piece of paper and assuming it is the unit of x and y. The space curve, Distance, Tangent, Normal, (& BiNormal in 3d), and curvature are then relative to and exist only on the piece of paper.

Because mathematicians teach math to engineers and physicists, the distinctions can become confusing. The physicists and engineers have the double burden of not only memorizing definitions, rules, and formulas, and being adept at their manipulation, but simultaneously creating some kind of real meaning to them in their minds. The result, sadly, is to cull out of potential engineers those who tend to think about the physical meaning of things, ie, potential engineers.