Is the direction derivatives of (orthonormal) normal vector(s) in its own direction in the tangent space?

Question

Let $M$ be a smooth manifold of dimension $d$ embedded in dimension $D>d$. Let $n_1,\dotsc, n_K$ be any orthonormal basis for $N_xM := T_xM^{\perp}$, the orthogonal complement of the tangent space of $M$ at $x$. Let $D[n_i]$ be the Jacobian of the vector field $n_i$ with respect to $x\in \mathbb{R}^D$. We assume that the $n_i$ are defined on a neighborhood of $M$ in $\mathbb{R}^D$.

Question: Is the vector $v_i(x) = D[n_i(x)]n_i(x)$ in the tangent space of $M$ at $x$?, i.e. $D[n_i]n_i \in T_xM$ for any $i$. (We are suppressing the notation $n_i(x)$ for $n_i$, out of laziness).

Partial Results: If $K=1$ the answer is yes. That is because we can easily prove for any $i$ that $n_i \perp v_i$ and if $K=1$ then there is only one normal spanning the orthogonal complement of the tangent space, so if we perpendicular to it, we must be tangent to the manifold, by definition of orthogonal complement.

For $K>1$, it eludes me. Here is what I have shown.

1. $n_j^T v_i = -n_i^T D[n_j]n_i$,

2. $n_i \perp D[n_i]u$ for any $u\in T_xM$, and

3. $n_j^T D[n_i]u = -n_i^T D[n_j]u$ for any $u\in T_xM$.

Possible counterexample: After thinking for some time, I began to suspect this is not true. Consider the space curve $\phi(x) = (x, x^2, 1-x)^T$. This has tangent $\phi'(x) = (1, 2x, -1)^T$. From which we can obtain normals via Gram-Schmidt, $$n_1 = [2x, -1, 0]^T/\sqrt{1+4x^2}$$ and $$n_2 = [3x^2, 6x^3, -(1+4x^2)]^T/\sqrt{(1+4x^2)(1+4x^2+9x^4)}.$$ And Jacobians $$D[n_1] = \begin{bmatrix} 2h(x)^{-3} & 0 & 0\\ 4xh(x)^{-3} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix},$$ where $h(x) = \sqrt{1+4x^2}$. It follows that $$n_2^T D[n_1]n_1 = \frac{12x^3}{(1+4x^2)\sqrt{1+8x^2+25x^4+36x^6}}\neq 0$$ for $x\neq 0$.

So if this is sound, then the question becomes: well when is it true, because it holds for, say, $\phi_1(x)=(x, 3x, \sin(x))^T$ and $(x,3x, x^3)^T$, but not $\phi_2(x) = (x, 3x, e^x)^T$.

I'm a beginner in anything related to differential geometry, so I apologize if there is something basic I am missing. I have found it one of the harder subjects to self-study compared to others I have tried.