The double point locus has the expected dimension $2m-n$

The map $f\times f$ is smooth on the open submanifold $M\times M\setminus {\mathrm{\Delta }}_M$ and transverse to ${\mathrm{\Delta }}_X$ there by [F1]; by [L3] the diagonal ${\mathrm{\Delta }}_X$ is an embedded submanifold of $X\times X$ of codimension $2n-n=n$. Applying [L1] to the restriction of $f\times f$ to $M\times M\setminus {\mathrm{\Delta }}_M$ yields that ${\mathrm{\Delta }}_2(f)$ is an embedded submanifold of $M\times M\setminus {\mathrm{\Delta }}_M$ of codimension $n$, that is, of pure dimension $2m-n$ when $2m-n\ge 0$, and in particular ${\mathrm{\Delta }}_2(f)$ carries the subspace topology by [L5].

The tangent space description at a point $(x,y)\in {\mathrm{\Delta }}_2(f)$ with $r=f(x)=f(y)$: by [L1], a tangent vector lies in $T_{(x,y)}{\mathrm{\Delta }}_2(f)$ exactly when its image under $d(f\times f{)}_{(x,y)}$ lies in $T_{(r,r)}{\mathrm{\Delta }}_X$. By [L6] applied to the components ${\pi }_1\circ (f\times f)=f\circ {\pi }_1$ and ${\pi }_2\circ (f\times f)=f\circ {\pi }_2$, this image is $(d{f}_x(v),d{f}_y(w))$ for a tangent vector $(v,w)\in T_{x}M\oplus T_{y}M$, while $T_{(r,r)}{\mathrm{\Delta }}_X=\{(u,u):u\in T_{r}X\}$; hence $T_{(x,y)}{\mathrm{\Delta }}_2(f)=\{(v,w)\in T_{x}M\oplus T_{y}M:d{f}_x(v)=d{f}_y(w)\}\mathrm{.}$

The set ${\mathrm{\Delta }}_2(f)$ is closed in $M\times M\setminus {\mathrm{\Delta }}_M$: it is the preimage under the continuous map $f\times f$ of the closed set ${\mathrm{\Delta }}_X$ by [L3] and [L9].

Suppose $2m-n<0$, that is $2m+n<2n$. The map $f\times f$, restricted to $M\times M\setminus {\mathrm{\Delta }}_M$, and the inclusion ${\mathrm{\Delta }}_X\hookrightarrow X\times X$ are transverse by [F1] and [L2], with source dimensions $2m$ and $n$ and target dimension $2n$; by [L4] their fibre product is empty. The fibre product projects bijectively onto the set of pairs $(u,\delta )$ with $f\times f(u)=\delta$ and $\delta \in {\mathrm{\Delta }}_X$, which is exactly ${\mathrm{\Delta }}_2(f)$, so ${\mathrm{\Delta }}_2(f)=\mathrm{\varnothing }$ and hence $\mathrm{\Sigma }(f)=f({\mathrm{pr}}_1{\mathrm{\Delta }}_2(f))=\mathrm{\varnothing }$.

The swap $\tau (x,y)=(y,x)$ is a diffeomorphism of $M\times M$ (an involution, smooth with smooth inverse), and it preserves $M\times M\setminus {\mathrm{\Delta }}_M$ and the condition $f(x)=f(y)$; hence it restricts to a smooth involution of the embedded submanifold ${\mathrm{\Delta }}_2(f)$ that is free, since $\tau (x,y)=(x,y)$ would force $x=y$, which is excluded on $M\times M\setminus {\mathrm{\Delta }}_M$.

The map $q(x,y)=f(x)=f(y)$ is smooth on ${\mathrm{\Delta }}_2(f)$: it is the restriction of the smooth ambient map $M\times M\setminus {\mathrm{\Delta }}_M\to X$, $(x,y)\mapsto f(x)$, so the criterion of [L7] applies. Its differential is injective at every point: by [L6], $d{q}_{(x,y)}(v,w)=d{f}_x(v)$ for $(v,w)\in T_{(x,y)}{\mathrm{\Delta }}_2(f)$; if $d{q}_{(x,y)}(v,w)=0$ then $d{f}_y(w)=d{f}_x(v)=0$ by the description of step 1.2, so $w=0$ and then $v=0$ because $d{f}_x$ and $d{f}_y$ are injective. Hence $q$ is an immersion, and by [L8] every point of ${\mathrm{\Delta }}_2(f)$ has a neighbourhood on which $q$ is an embedding onto an embedded submanifold of $X$.

By definition of $\mathrm{\Sigma }(f)$ as $f({\mathrm{pr}}_1{\mathrm{\Delta }}_2(f))$ the image of $q$ is $\mathrm{\Sigma }(f)$, and $q\circ \tau =q$ because $\tau$ only exchanges the two coordinates, which have equal images.

The fibres of $q$ are unions of $\tau$-orbits: $q(x,y)=r$ if and only if $x$ and $y$ are two distinct points of the preimage $f^{-1}(r)$, so $q^{-1}(r)$ is in bijection with the ordered pairs of distinct points of $f^{-1}(r)$, on which $\tau$ acts by exchanging the two entries. Consequently $q$ induces a well-defined surjection ${\mathrm{\Delta }}_2(f)\mathrm{/}\tau \to \mathrm{\Sigma }(f)$ sending the orbit of $(x,y)$ to $f(x)$, and this map is injective exactly when every fibre $f^{-1}(r)$ with $r\in \mathrm{\Sigma }(f)$ consists of exactly two points, that is, exactly when no point of $X$ is the image of more than two points of $M$.

The claims are steps 1.1 and 1.3 for clause 1, step 2.1 for clause 2, and steps 2.2, 2.3, 3.1 and 4.1 for clause 3. No finiteness of ${\mathrm{\Delta }}_2(f)$ was used or asserted.