A finite morphism $f:X\rightarrow Y$ firstly requires an affine cover $V_i\subset Y$, such that $f^{-1}(V_i)$ are affine open sets. However a morphism of (locally) finite type $f:X\rightarrow Y$ involves a cover $V_i\subset Y$ such that $f^{-1}(V_i)$ can be covered by affine open sets.
The latter sense of local must be broader than the first one. Why are the first part of their definition so different? What if I define a finite morphism by requiring the preimage covered by affine open sets, such that the induced $f^*$ are integral, will undesirable things happen? (BTW, is it right that there's actually no difference when $X$ is affine?)