Embedding, local diffeomorphism, and local immersion theorem. Suppose $f: M \to N$ is smooth and an immersion, i.e $df_p : T_p(M) \to T_p(N)$ is one-to-one. Since $f$ is an immersion, we have the following theorem, 
$\textbf{Local Immersion Theorem:}$ Suppose that $f: M \to N$ is an immersion at $x$. Let $y=f(x)$. Then there exists local coordinates around $x$ and $y$ such that $$ f(x_1, x_2, \dots, x_k) = (x_1, x_2, \dots, x_k, 0, \dots, 0 )$$
In other words, $f$ is a locally one-to-one, and thus an embedding locally. Does this imply that $f$ is a local diffeomorphism?
I am looking for a answer as to the relationship between the three concepts: local immersion theorem, local embedding, and local diffeomorphism. 
I know some similar questions have been asked, but in more specific circumstances

Heavens, no! The differential $df$ maps from a $k$-dimensional vector space to an $n>k$ dimensional vector space. It cannot be an isomorphism.
However, by the local coordinates condition you've imposed, the differential is full-rank, and so $f$ is a local diffeomorphism onto its image.
