The essence of this question is:
Let $(M,g_M)$ and $(N,g_N)$ be Riemannian manifolds. How many different ways are there to embed $M$ isometrically in $N$?
In this context, I say the embedding $i_1$ is different from $i_2$ if there is no isometry $\varphi:N\to N$ suct that $i_2=\varphi\circ i_1$.
This question is rather easy for specific pairs of manifolds. For example, if $M$ is a line, any embedding has an arc length parametrization, which is just a different name for isometric embedding. Alternatively, let $M$ be Euclidean plane and $N$ Euclidean space. Then for every (smooth) function $f:\mathbb{R}\to\mathbb{R}$, the submanifold $$\{(x,y,z)|y=f(x)\}\subset\mathbb{R}^3,$$equipped with the Riemannian metric inherited from $\mathbb{R}^3$, is isometric to the plane. It is not hard to convince oneself that many of these submanifolds cannot be carried to one another by isometries of Euclidean space.
However, there are other simple examples for which I don't know the answer. For example, what happens when $M$ is $S^2$ with the unit sphere metric, and $N$ is Euclidean space? Somehow, the only way I can imagine $S^2$ embedded in $\mathbb{R}^3$ is by the standard embedding, but I can't find an argument to prove that. Intuitively speaking, it may have something to do with curvature. If so, I guess the answer to the question changes when allowing $C^1$ embeddings.
Anything regarding the last example, of the sphere and Euclidean space, would already make me happy. If, by any chance, someone has an answer to a more general case, that would be great. Insights about less smooth embeddings (e.g. $C^1$ embeddings) are also welcome, though I'm mostly interested in the smooth case.