Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set.

In Evans book we find the definition

$$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on bounded subsets of} U, \text{ for all }|\alpha|\leq k \}$$

it is easy to show that this is equivalent to

$$ C^k(\overline U) = \{f\in C^k(U)| D^\alpha f \text{ can be extended continuously on } \overline U, \text{ for all } \}$$

In differential geometry the definition

$$ C^k(\overline U) = \{f|_{\overline U} \colon \exists O\supset \overline U \text{ such that } f\in C^k(O)\}$$

Is it obvious that the first two definitions are equivalent to the last one?

Edit: Or was this identity first shown in Whitney’s extension theorem?

Edit 2:
Conisdering Guiseppe Negro’s comment: Since the last definition is used in differential geometry, is it true that those definitions are equivalent when the set $\overline U$ is a manifold or manifold with boundary?

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