Notions of equivalent metrics
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and $d'$, in increasing order of strength:
$i$ is a homeomorphism, i.e. $d$ and $d'$ induce the same topology on $X$.
$i$ and $i^{-1}$ are uniformly continuous.
$i$ is bilipschitz, i.e. $C_1 d' \le d \le C_2 d'$.
I would like to know what terms are used for these notions.
In particular, Mathworld says that the term "equivalent" refers to sense 1. This seems counterintuitive since, for instance, sense 1 does not preserve completeness. Munkres's General Topology uses "metrically equivalent" for sense 2. Does this agree with people's experience of standard usage?
Edit: I will point out that 3 implies 2 implies 1 (since Lipschitz implies uniformly continuous implies continuous) but converses are false. Let $X = \mathbb{R}$, let $d_1(x,y) = |x-y|$, $d_2(x,y) = |x-y| \wedge 1$, $d_3(x,y) = |\phi(x)-\phi(y)|$, where $\phi : \mathbb{R}\to (0,1)$ is your favorite homeomorphism. In sense 1 all three are equivalent, in sense 2 $d_1 \sim d_2 \not\sim d_3$, and in sense 3 all are inequivalent. In particular note that $d_1, d_2$ are complete but $d_3$ is not.
Edit: Further confusing the matter is the fact that if $X$ is a vector space and $d,d'$ are induced by norms $||\cdot||, ||\cdot||'$, then all three senses coincide, and sense 3 is usually taken as the definition of "equivalent norms."
I don't know if there is a standard, but I can provide two sample points.
Burago, Burago, and Ivanov in "A course in metric geometry", p. 9 call definition (3) Lipschitz equivalent.
Dugundji in "Topology" calls those metrics satisfying (1) equivalent.
Burago et al. is available online.
The first one is simply the definition of topological equivalence - it verbatim extends to general topological spaces (not necessarily metric spaces). I don't have a name for the second one. The third, which is used often, is called "Lipschitz equivalence".