Understanding equivalent metric spaces
Solution 1:
The short answer to "Is there any connection between homeomorphism and equivalence of metric spaces?" is yes. The long answer: any reasonable notion of equivalence of two metrics $d_1$ and $d_2$ can be formulated in terms of the identity map $\mathrm{id}\colon (X,d_1)\to (X,d_2)$. As soon as we distinguish a class of "nice" maps (the class should be a group under composition), we get a notion of equivalence. Some examples, previously mentioned and not:
$\mathrm{id}$ is a homeomorphism
$\mathrm{id}$ is a uniform homeomorphism (i.e., uniformly continuous with a uniformly continuous inverse)
$\mathrm{id}$ is a bilipschitz homeomorphism (i.e., Lipschitz with a Lipschitz inverse)
$\mathrm{id}$ is a quasisymmetric homeomorphism
$\mathrm{id}$ is an isometry
$\mathrm{id}$ is a quasi-isometry
... The list is not exhaustive.
The corresponding notions of equivalence are related by $5\implies 3\implies 2\implies 1$, also $3\implies 4\implies 1$, and $3\implies 6$.
Solution 2:
Two definitions on equivalence are not the same. Definition 2 implies definition 1, but not vice versa.
For instance, Let $X = (0, 1]$ and $d_1(x, y) = |x - y|$ and $d_2(x, y) = |\frac 1 x - \frac 1 y|$. Then $d_1$ is equivalent to $d_2$ under Def-1, but not under Def-2. Indeed, one can see $X$ is complete under $d_1$, but not $d_2$.