I think what you need is the following lemma (usually called the "Stack of records" lemma):

Consider a smooth proper map of manifolds of the same dimension $f \colon M \to N$ and let $y \in N$ be a regular value of $f$.

Then there exists a neighbourhood $V \subset N$ of $y$ such that $f^{-1}(V) = \cup_i cup_{i=1}^n U_i$ with $U_i \cap U_j = \emptyset$ for $i \neq j$ and $f|_{U_i} \colon U_i \to V$ is a diffeomorphism for all $i$.

Now from this you can just sum up $\pm 1$ according to orientation on each $U_i$ to get the local degree of $f$ at $y$, and this works for both definitions of degree.

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I think what you need is the following lemma (usually called the "Stack of records" lemma):

Consider a smooth proper map of manifolds $f \colon M \to N$ and let $y \in N$ be a regular value of $f$.

Then there exists a neighbourhood $V \subset N$ of $y$ such that $f^{-1}(V) = \cup_i U_i$ with $U_i \cap U_j = \emptyset$ for $i \neq j$ and $f|_{U_i} \colon U_i \to V$ is a diffeomorphism for all $i$.

Now from this you can just sum up $\pm 1$ according to orientation on each $U_i$ to get the local degree of $f$ at $y$, and this works for both definitions of degree.