While the OP is pondering which interpretation of the question is the right one, here are some relevant references. The characterization of compactness in the little Hölder space given by Giorgio Metafune is Thm.3.2 in (Johnson, 1970).
As for the big Hölder and Lipschitz spaces, a 2019 monograph dedicated to these function spaces does not seem to contain an analog of an Arzelà-Ascoli-like characterization of compact subsets. So that could still be an open problem.
Update: Some incidental searching of the literature uncovered this nice (and essentially hot of the presses) article:
Gulgowski, Jacek; Kasprzak, Piotr; Maćkowiak, Piotr, Compactness in Lipschitz spaces and around, ZBL07762688, arXiv:2205.04543.
Let me quote from their Introduction, which confirms that there was a gap in the literature concerning compactness characterizations in Lipschitz and Hölder spaces:
Because of their applications—[...]—Lipschitz and Hölder continuous functions constantly enjoy great interest among researchers. Each year numerous scientific articles and chapters are devoted just to studying their properties. Even whole books on Lipschitz and Hölder functions are written. [...] What they lack, however, is a (strong) compactness criterion in such spaces. No matter how hard we tried, we could not find one in the literature, although, as it turned out, in the case of real-valued functions it had stayed hidden in a plain
sight for more than half a century. In a paper of Cobzaş from 2001 it is even stated: “[...] apparently there is no compactness criterion in spaces of Hölder functions, and some criteria given in the
literature turned to be false” (see [8, p. 9]).
In the body of the article they fill this void with a mixture of old and recent ideas. The recent ideas are really only needed to state the results in the generality that they want, which is for functions valued in Banach spaces. For scalar valued functions old ideas are are actually sufficient, which is kind of obvious in retrospect.
The key idea relies on the following observation. Let $(X,d)$ be a metric space and denote by $\def\Lip{\operatorname{Lip}} \Lip_\alpha(X)$ the space of $\alpha$-Lipschitz/Hölder continuous scalar valued functions on $X$ ($0 < \alpha \le 1$, with $\alpha=1$ corresponding to the Lipschitz case), as well as $\Delta_X = \{(x,x) \mid x \in X\}$, $X^{(2)} = X \times X \setminus \Delta_X$.
Lemma: The ( so-called de Leeuw's ) map
$$
\Phi\colon \Lip_\alpha(X) \to C(X^{(2)}), \quad
\Phi(f)(x,y) = \frac{f(x) - f(y)}{d(x,y)^\alpha}
$$
has only constants in its kernel, has a closed range and is a homeomorphism of $\Lip_\alpha(X) / \{\text{const}\}$ onto its image.
Proof: Mostly self-evident, taking into account the form of the Hölder norm and the $\sup$ norm on $C(X^{(2)})$. It is worth noting that the range is closed, since it can be realized as the kernel of a continuous linear map, for instance $\Psi \colon C(X^{(2)}) \to C(X^{(2)}) \times C(X^{(3)})$, where $X^{(3)} = X \times X \times X \setminus (\Delta_x \times X \cup X \times \Delta_X)$ and $$\Psi(g) = (g(x,y)+g(y,x), ~~ d(x,y)^\alpha g(x,y) + d(y,z)^\alpha g(y,z) - d(x,z)^\alpha g(x,z)). \tag*{$\Box$} $$
The statement of the Lemma is deliberately vague. There are different versions of Lipschitz spaces (local, global, global and uniformly bounded), which can be accommodated by adjusting $C(X^{(2)})$ and the topology on it. The simplest case is just to take uniformly bounded continuous functions on $X^{(2)}$ under the uniform norm, corresponding to uniformly bounded Lipschitz functions on $X$. For locally Lipschitz functions, one should take a filter of neighborhoods of the missing $\Delta_X$ rather than all of $X^{(2)}$.
Finally, the desired characterization is provided by the now obvious
Proposition: Compactness of $A \subset \Lip_\alpha/\{\text{const}\}$ is characterized requiring boundedness of $|f(x)|$ uniformly over $f\in A$ (for any one point $x\in X$) and by applying the appropriate version of the Arzelà-Ascoli theorem in $C(X^{(2)})$ to the image $\Phi(A)$ of de Leeuw's map.
In addition: When $X \subseteq \mathbb{R}^n$ is (at least) a sufficiently nice subset, there is another approach. Namely, via a combination of Rademacher's theorem and Morrey's inequality, we have (e.g., Thm.5.8.4 in Evans PDEs, or Thm.1.41 of Weaver Lipschitz functions) a linear isometry $\Lip_1(X) \cong W^{1,\infty}(X)$, which is the first order Sobolev space consisting of bounded $f$ such that $df \in L^\infty(X)^n$ with respect to Lebesgue measure. By definition of the Sobolev space, the exterior derivative is a linear isometry $d \colon W^{1,\infty}(X) \cong L^\infty(X)^n$.
So, as earlier, compactness of $A\subset \Lip_1(X)$ can be characterized by equiboundedness at a point and compactness of $d(A)$, while compactness in $L^\infty$ can be characterized using Thm.IV.8.18 (or adapting Thm.IV.5.6) of Dunford & Schwartz, vol.I.
I've not seen anything similar for Hölder spaces, though.
