Different definitions of Novikov ring?

If you read around, you'll find plenty of other variants of the Novikov ring... The underlying point is that to apply the Gromov compactness theorem (or its algebraic counterpart) you need a bound on the energy (= symplectic area) of ridid holomorphic spheres. Since there's no a priori bound in general, you instead count curves according to their area, or according to some refinement thereof, such as homology class.

Your idea is correct: if we fix a base field $k$, the homomorphism $H_2(X)/tors. \to \mathbb{R}$ given by pairing with $[\omega]$ induces a homomorphism from the completed group $k$-algebra on $H_2(X)/tors.$ to the "universal Novikov field" (that is, the field of series $\sum_{\lambda \in \mathbb{R}}{a(\lambda)T^\lambda}$ where the support of $a \colon \mathbb{R}\to k$ contains only finitely many reals less than any given $C$). After making a small perturbation to $\omega$, changing its cohomology class, we can make this homomorphism injective. If memory serves, the smaller Novikov ring is a PID, and the universal Novikov field is flat over it because it's a torsion-free module over a PID. So you can safely tensor with it after passing to cohomology.

The advantages of the universal Novikov field are that it's universal (meaning biggest, essentially), that it's a field, and that it comes with a natural valuation. The disadvantage is that it's bigger than is strictly necessary.

We must think of $T$, the generator of the Novikov ring, as a coordinate on the moduli space of symplectic structures $\mathcal M_{sym}(X)$. We can define $<\cdots>_g^\psi:=\sum_\beta<\cdots>_{g,\beta}^\psi e^{-\omega(\beta)}$ , but we don't know if the right hand side converges and so we put $T^{\omega(\beta)}=e^{-(-\log T\omega)(\beta)}$ and we can think of this as large volume limit, i.e. neighborhood of singular point in $\mathcal M_{sym}(X)$ where $\omega\to \infty$

For a $J$-holomorphic curve $u$ we have $E(u)=ω([u])=∫_Σu^∗ω≥0$, and we must take $λ_i≥0$. Moreover if $ω$ belongs to an integral cohomology class, then $ω(β)∈\mathbb Z_{≥0}$ and we can replace the Novkov ring with $\mathbb Q[[T]]$

See Denis Auroux's paper https://arxiv.org/pdf/0902.1595.pdf .