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So for the definition of Dehn surgery (also called rational/integer surgery), what is correct:

Definition Let $K$ be a knot in an oriented $3$-manifold with a regular neighbourhood $N(K)\simeq S^1\times D^2$. Dehn surgery is the operation of removing $\operatorname{int} N(K)$ and gluing in $S^1\times D^2$ by an arbitrary / orientation reversing (?) diffeomorphism mapping from the boundary torus $S^1\times \partial D^2$ to the boundary of $N(K)$.

I am a little undecided after scanning the literature.

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It doesn't matter.

Suppose $M$ is a 3-manifold with torus boundary and $N = S^1 \times D^2$ is the solid torus. Then if $\varphi,\psi: \partial M \to \partial N$ are diffeomorphisms, if $\psi \circ \varphi^{-1}$ extends to a diffeomorphism of $N$, then $M \cup_\varphi N \cong M \cup_{\psi} N$.

Now if $\varphi$ is an orientation-preserving (or reversing) diffeomorphism, and $r$ is the diffeomorphism represented by the matrix $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

then $\varphi \circ r$ is an orientation-reserving (or preserving) diffeomorphism, and $M_\varphi N \cong M_{\varphi r} N$, because $r$ extends over $N$.

So you get the same manifolds no matter whether you demand some orientation restriction or not.

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