I presume you got these equations from a derivation similar to https://en.wikipedia.org/wiki/Deformation_(mechanics)#Normal_strain
For small (infinitesimal) strains, the second order (squared) terms are negligible compared with $\partial u/\partial x$, and are simply ignored. This gives what is commonly called "engineering strain".
For large strains, you need a more careful definition of "strain" that gives zero strain for arbitrarily large rigid body rotations of a small element of material. See https://en.wikipedia.org/wiki/Finite_strain_theory for details. The Cauchy-Green strain tensor is commonly used to model situations where small elastic deformations of the material are superimposed on large rigid body translations and rotations.
As a general comment, there is no single "theoretically correct" definition of strain. If you assume the material properties (Young's modulus, Poisson's ratio, etc) are constant in the stress-strain equations, you will get different results depending on which strain measure you take - but in real life, the material properties are strain-dependent in any case, so you can make an accurate (but nonlinear) model of the material with any reasonable definition of strain.