What does it mean for two submanifolds to have contact of order $k$? Let $M$ be a smooth manifold of dimension $(m+n)$. Two curves $\gamma_1, \gamma_2 \colon \mathbf{R} \to M$ with $\gamma(0) = p$ are said to have contact at $p$ of order $k$ if for all smooth maps $\varphi \colon U \to R$ where $U \subset M$ is open about $p$, we have the following equality of jets: $J_0^k(\varphi\gamma_1) = J_0^k(\varphi\gamma_2)$. 
Consider now submanifolds $N_1$ and $N_2$ of codimension $m$. How one would define $k$-th order contact at $p \in N_1 \cap N_2$? 
EDIT: While we wait for Ted's reply, I find this definition to be reasonable:
$N_1$ and $N_2$ have $k$-th order contact at $p \in N_1 \cap N_2$ if for all $[\gamma]$ in the jet space $J_0^k(\mathbf{R}, M)_p$ there are submanifoldcharts $(x, U)$ and $(y, V)$ for $N_1$ and $N_2$ respectively such that $J_0^k(x\gamma) = J_0^k(y\gamma)$.
 A: OK, so I want to define the $k$th order osculating space at $p$ of the submanifold $N$ to be 
$$T^{(k)}_p(N) = \text{Span}\big(\frac{\partial f}{\partial x_i}(0),\frac{\partial^2 f}{\partial x_i\partial x_j}(0), \dots, \frac{\partial^k f}{\partial x_{i_1}\dots\partial x_{i_k}}(0)\big),$$
where I'm assuming (having chosen a chart for $M$) that $f\colon U\to\Bbb R^{n+m}$ is a local parametrization of $N$ ($0\in U\subset\Bbb R^n$, $f(0)=p$).
You can check, using the chain rule, that this is a well-defined subspace of $T_pM$. 
Indeed, that calculation shows precisely that the definition you wished to use is equivalent. If $g=f\circ\phi$ for some diffeomorphism $\phi\colon U\to V$ with $\phi(0)=0$, then you can show inductively that
$$D^ig(0) = \sum_{j=1}^i c_{jA} D^jf(0)(D^{\alpha_i}\phi(0),\dots, D^{\alpha_j}\phi(0))$$
for appropriate constants $c_{jA}$, where we sum over increasing multiindices $A$ with $\alpha_1+\dots+\alpha_j = i$. (Here I'm treating $D^jf$ as a symmetric $j$-linear map.)
Conversely, because we know that $Df(0)$ has maximal rank, we can also deduce that if the $k$th order osculating spaces according to $f$ and $g$ at $p$ are equal, then we can construct a polynomial $\phi$ of degree $k$ so that $g=f\circ\phi$ (on appropriate domains).
