Mark gives a good answer. I thought to make this a comment on his answer, but I rambled on past the allowed length and so post as an answer.
It is all a matter of what maps one wants to study. As Mark noted, most analysts as well as probabilists are most interested in (1), the category Ban. The less flexible and easier to treat category Ban$_1$, given by (2), is interesting for many people and was the first to be developed to a high degree. PDE people, interested in Lipschitz mappings, naturally are care about biLipschitz equivalence: some geometers like uniform equivalence; while geometric group theorists are mostly interested in coarse equivalence. I am interested in all of these notions of equivalence and more.
No matter what category one works in, the word isomorphism means linear homeomorphism (usually into; one adds onto or surjective when called for). Other notions of being the same are called isometric, Lipschitz equivalent, uniformly equivalent, coarsely equivalent. For a time, geometric group theorists called "coarse equivalence" "uniform equivalence", but this fortunately is passing.
From a Banach space theoretic perspective, one major challenge is to determine when a weaker notion of equivalence (or embedding) implies isomorphic equivalence (or isomorphic embedding). This is interesting also for people who use Banach spaces without doing Banach space theory. Take, for example, geometric group theorists. Yu and then Kasparov and Yu proved numerous results about finitely generated groups whose Cayley graphs coarsley embed into a "nice" Banach space. For a time it was open whether every "nice" (in this case uniformly convex) Banach space embeds into a Hilbert space--were this true, they could have ignored other Banach spaces. Alas (or YES!, depending on your point of view), that is not the case. It is now a research topic of interest to a large group to determine when a Banach spaces embeds coarsely into a special Banach space $X$. For $X$ a Hilbert space (or, more generally, $L_p$ and $\ell_p$ for $p \le 2$), the answer was provided by Nirina Randrianarivony, but there are only partial results for $L_p$ when $2<p<\infty$.