In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition for the existence of an extension operator (for Whitney jets on compact $K\subset \mathbb{R}^n$ to $\mathbb{R}^n$; if $K$ is the closure of its interior I gather this is the same as smooth functions on $K$) is given:

Assume there exist $\varepsilon_0,\ \rho\gt 0$, $r\geq 1$ such that for all $z$ in the boundary of $K$ and $0\lt \varepsilon \lt\varepsilon_0$ there is $x\in K$ with $|x-z|\lt\varepsilon$ and $\{y: |y-x| \leq \rho\varepsilon^r\} \subset K$. Then $\mathcal{E}(K)$ satisfies (DN) and therefore it admits an extension operator.

This implies the version proved much earlier by Stein that covers the case of $Lip_1$ or $C^1$ boundary, or convex sets with non-empty interior.

I'm interested in the case where I don't just take compact sets in Euclidean space, but some more general (in fact compact) manifold $M$, possibly with boundary (or even corners). I certainly am only interested in sets $K$ that are closures of open sets, so (as far as I can tell) I'm really working with smooth functions here.

For our general compact manifold-with-boundary $M$ pick a metric $g$, giving $M$ bounded geometry. (Aside: I believe one could even find a real-analytic structure on $M$, see e.g. K. Shiga, Some aspects of real-analytic manifolds and differentiable manifolds J. Math. Soc. Japan Volume 16, Number 2 (1964), 128-142. However I'm suspicious about the existence of a real analytic metric. Once one exists, then certainly one exists giving $M$ bounded geometry by ibid.)

So I am interested to know whether given some $K\subset M$ contained in a single chart, hence diffeomorphic to $K' \subset \mathbb{R}^n$, the two ways of defining a topology on what one might call $\mathcal{E}(K)$ coincide. That is, consider $\mathcal{E}(K')$ as defined in eg Frerick's paper linked above, or consider a variant of the topology defined directly from $K$ using the chosen metric $g$ for estimates. Clearly this reduces to the case where we have transported $g$ across the chart map, so that we are reduced to considering the case of $K'$, where now $\mathbb{R}^n$ is given some metric $g'$ (and one can choose it to be of bounded geometry, as the closure of the chart in $M$ is compact). Call this space $\mathcal{E}_{g'}(K')$.

Now I'm interested in compact sets that are closures of finite intersections of geodesically convex balls in general position, hence closures of sets that are geodesically convex. If I can believe that $\mathcal{E}(K') \simeq \mathcal{E}_{g'}(K')$ as Fréchet spaces, then I would like to apply an analogue of Frerick's result, using the metric induced by $g'$ instead of the usual norm. Then, relying on geodesic convexity of my set of interest, I hope to conclude that an extension operator exists.

Does this argument pan out?