In David Mumford's "Red Book of Varieties and Schemes" in Lecture IV: The Torelli Theorem and the Schottky Problem of the Appendix on page 271 is described a condition when for $g \ge 2$ and lattice $L \subset \mathbb{C}^g$ the complex torus $X= \mathbb{C}^g/L$ becomes projective variety. The necessary and sufficient ingredient is the existence of a positive definite Hermitian form $H$ on $ \mathbb{C}^g $ such that $E:= Im (H)$ is integral on $L \times L$. Such complex tori $X$ are called abelian varieties. The forms $H$ are called polarizations of $X$.
Questions:
A By definition $H: \mathbb{C}^g \times \mathbb{C}^g \to \mathbb{C}$. What does it mean that "$E:= Im (H)$ is integral on $L \times L$"? Does this conditions literally say that the image $H(L \times L) \subset \mathbb{C}$ is contained in $\mathbb{Z} \subsetneqq \mathbb{C}$? If not, what is the meaning of the quoted sentence?
B About polarizations in general. The text says "The forms $H$ are called polarizations of $X$". So literally a "polarization" can be defined independ of theory of algebraic varieties with pure linear algebra methods as a positive definite Hermitian bilinear form on a complex vector space $\mathbb{C}^n$, as far right?
Moreover the name "polarization" sounds very geometrical. Is there any way how to think about such polarization? Does there exist probably any archetypical example that motivates & justifies the naming "polarization"?