I'm studying calculus and our professor gave us two definitions of manifolds:
$1) M\subset{R^n}$ is a k-dimensional manifold if for every $x\in M$ there exists a neighborhood $U_x\subset R^n$ such that $M\cap U_x$ is a graph, i.e there exist indexes $i_1,...,i_k$ and $j_1,...j_{n-k}$ which their union is $\{1...n\}$ and an open subset $V\subset R^k$ and a smooth function $f:V\to R^{n-k}$ such that $M\cap U_x=\{x\in R^n | (x_{j_1},...,x_{j_{n-k}})=f(x_{i_1},...x_{i_k}), (x_{i_1},...x_{i_k})\in V\}$.
$2)M\subset{R^n}$ is a k-dimensional manifold if for every $x\in M$ there exist a neighborhood $U_x$ and a diffeomorphism $\phi:U\to \phi(U)$ such that $\phi(M\cap U)=\phi(U)\cap (R^{n-k}\times\{0\}_{n-k})$ when $\{0\}_{n-k}$ represents the n-k dimensional $0$-vector.
The first thing I'm having trouble is understanding the meaning of the second definition. I miss some intuition to why it is defined this way. The second thing is understanding why the two definitions are equivalent.