What is an integral differential form and how do we recognize it as such?

Question

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be endowed with an integral 2-form. What does this mean?

Let me take a chance at guessing an answer: a $p$-form is integral if its pairing against any $p$-chain is an integer. Alternatively, the $p$-form should be in the image of the embedding of $H^p _{simplicial} (M, \Bbb Z)$ into $H^p _{deRham} (M, \Bbb R)$ (would anything change if I complexified the manifold and used $\Bbb C$ instead of $\Bbb R$?) Is my guessing correct?

Checking whether a form is integral seems very complicated using the definition. Are there more humane sufficient conditions implying this integrality test (I'm mostly interested in symplectic forms)?

EDIT:

Let me try to resuscitate this, because I am still confused.

So far, I have seen three ways of defining the integrality of the cohomology class $[\omega]$ of a closed $2$-form $\omega$:

1. by requiring $[\omega]$ to belong to the image of $H^2 (M, \Bbb Z)$ in $H^2 (M, \Bbb R)$; this is the most concise but also the most abstract and difficult to work with version;

2. by requiring that $\int _S \omega \in \Bbb Z$ for all closed oriented surfaces $S$ embedded in $M$;

3. the most technical of them all: for any cover $M$ with (contractible? is this necessary?) open sets $U_i$ on which $\omega = \textrm d \theta_i$; on $U_i \cap U_j$ we have $\textrm (\theta_i - \theta_j) = \textrm d \textrm d \omega = 0$, so $\theta_i - \theta_j = \textrm d f_{ij}$; note that on $U_i \cap U_j \cap U_k$ we have $\textrm d (f_{ij} + f_{jk} + f_{ki}) = \theta_i - \theta_j + \theta_j - \theta_k + \theta_k - \theta_i = 0$, so $f_{ij} + f_{jk} + f_{ki} = c_{ijk} \in \Bbb R$; $[\omega]$ is said to be integral if and only if $c_{ijk} \in \Bbb Z \ \forall i,j,k$.

My problem is that I fail to understand whether the three of them are equivalent, and if so - why. For instance, (2) is almost an immediate corollary of (1), but does (2) imply (1)? Why may I test integrality by using just embedded smooth oriented surfaces, instead of general cycles from $Z_2 (M, \Bbb Z)$? Number (3)'s connection with the others baffles me.