I am reading parts of Mumford's Red book of Varieties and Schemes https://link.springer.com/book/10.1007/b62130. In Chapter III Section 2, in Definition 5 he defines a vector bundle of rank $r$ with atlas. Right after the definition, he writes: The simplest way to define a vector bundle, without distinguishing one "atlas", is to simply add the extra condition that the atlas is maximal, i.e. every possible open $V \subset X$ and isomorphism $\phi: \pi^{-1}(V)$ to $\mathbb{A}^r \times V$ compatible with the given $(U_i, \phi_i)$ is already there.
I am confused with what he means by "without distinguishing one atlas" and what this is about, which is probably due to my lack of background in differential geometry. Any clarification is appreciated. Thank you.
A manifold can have many different atlases. For example the sphere can be covered by $\Bbb S^2\setminus \{n\}$ and $\Bbb S^2\setminus\{s\}$ (where $n,s$ denote the north/south pole respectively), but you can also cover it by smaller caps, like north-western, north-eastern,... Thus to avoid always having to state what atlas to use one simply uses all at once, that is a maximal atlas.
I guess the story with vector bundles is the same.