It seems as a general rule of thumb it's safe to assume "$\alpha$-Hölder" means (possibly locally) $\alpha$-Hölder with $\alpha\in]0,1[$; Lipschitz functions are qualitatively much stronger than $\alpha$-Hölder functions with $\alpha<1$ (e.g. in hyperbolic dynamics the stable and unstable distributions are automatically Hölder but not Lipschitz; see Why Do We Care About Hölder Continuity?), and existence of non-constant functions that are $\alpha$-Hölder with $\alpha>1$ indicates that the underlying metric geometry is subtler than Euclidean. For instance here is an excerpt from the appendix by Semmes to Gromov's Metric Structures for Riemannian and Non-Riemannian Spaces (p.423):
When $\alpha = 1$ we have already christened this the Lipschitz condition,
but the latter is special and deserves a special name. Traditionally one does
not consider $\alpha > 1$ because on many spaces Hölder continuous functions
of order $\alpha > 1$ are all constant. This happens on Euclidean spaces, for instance, because Hölder continuity of order $\alpha > 1$ implies that the first derivatives of the function exist and vanish everywhere.
Notice that if we replace $(M,d(x,y))$ with the snowflake $(M,d(x,y)^\alpha)$
then [the $\alpha$-Hölder condition for $d$] becomes simply the Lipschitz condition with respect to the new metric. So all theorems about Lipschitz functions on metric spaces apply to Hölder continuous functions. The reverse fails in a strong way, as we shall see.
Typically when there are plenty of functions which satisfy a condition like [the $\alpha$-Hölder condition for $d$] with $\alpha > 1$ it means that the space that we are working on is already the result of applying the snowflake functor to a metric space. One can formulate precise versions of this statement which are easy to prove.
To detect the range of $\alpha$ it might be useful to be mindful of where the convexity/concavity of $t\mapsto t^\alpha$ is used (if at all).
Regarding non-negative exponents; (for all its worth) I personally never saw the phrase "$0$-Hölder"; it's hard to imagine "$0$-Hölder" meaning anything other than bounded. Functions with negative exponents could be useful as "moduli of total discontinuity"; though it's unclear to me if such things are studied.
Note that there is a similar situation with $L^p$ spaces; typically $p\in[1,\infty[$, possibly $p\in[1,\infty]$, but at times $p\in]0,\infty]$ (or even $p=0$ too, denoting the space of measurable functions, as in Notation for the set of measurable functions and the related quotient space).
