@rubi I have found the source of our disagreement. We are both right, but we are using different definitions of "curve". In literature one can find two different definitions:
Definition 1: Let ##M## be a manifold. A curve is a
map ##\mathbb{R}\rightarrow M##.
Definition 2: Let ##M## be a manifold. A curve is the
image of a map ##\mathbb{R}\rightarrow M##.
According to Def. 2, a curve is a 1-dimensional submanifold of ##M##. The curve in Def. 2 is an equivalence class of all curves in Def. 1 with the same image on ##M##. In the book
Y. Choquet-Bruhat, C. DeWitt-Morette, Analysis, Manifolds and Physics 1 the first definition of curve is called the
parameterized curve, while the second definition of curve is called the
geometric curve.
For instance, if ##M## is a 2-dimensional manifold with coordinates ##x,y##, consider the parabola defined by
$$y=x^2$$
This parabola is not a curve according to Def. 1, but is a curve according to Def. 2. (To further complicate terminology let me also note that, in algebraic geometry, an object such as that parabola is called a
variety.)
Your claims are right if by "curve" one means the first definition. My claims are right if by "curve" one means the second definition. Neither of us presented the explicit definition of the curve because we were not aware that there are two inequivalent definitions. I think that this resolves our disagreement. So instead of my previous

, now my face looks more like

.