In the literature there are two different definitions of polarized K3 surfaces.
1) A polarized K3 is the data $(X,\omega)$. Where $X$ is a K3 surface and $\omega$ is an ample class in $H^2(X,\mathbb{Z})$.
2) A polarized K3 is the data of a K3 surface $X$ together with a polarization on the Hodge decomposition $H^2(X,\mathbb{Z})$, which is a pairing $H^2(X,\mathbb{Z}) \times H^2(X,\mathbb{Z}) \rightarrow \mathbb{Z}$. Such a polarization is given by the intersection pairing: $(v,w) \mapsto \int_X v \wedge w\;.$
This, together with an observation from Huybrechts lecture notes on K3 surfaces page 40:
...it is not the intersection pairing that defines a polarization, but the pairing that is obtained from it by changing the sign of the intersection pairing for an ample class.
justifies the following question.
Question: How does the choice of an ample class $\omega$ comes into the definition of the intersection pairing?
It is clear that we need such a class to define a polarization on lower degree cohomology groups, for example on $H^1(X,\mathbb{Z})$. And the well definition of the intersection pairing is related to the existence of an ample class (i.e. the projectivity of $X$). But I don't see how the intersection pairing would change changing the choice of the ample class.
It is possible (or even probable) that I'm making a huge mess out of nothing, but I'm quite confused. Thank you in advance!