Does the cancellation property for a group mean something different than the cancellation property for an integral domain?

If you look at the case of the integers $\mathbb Z$, you can prove that it has the cancellation property by using the fact that it has no zero divisors. However, many people would take an approach that is technically much more difficult by arguing that $\mathbb Z$ can be embedded (as a ring) in the field of rational numbers $\mathbb Q$, in which the cancellation property holds because of the existence of inverses. This other argument can also be extended to arbitrary integral domains, by considering the field of fractions of the integral domain.

Whether this makes the cancellation property fundamentally the same for integral domains and fields, I'll leave for you to judge. I think the important point is that, even if such constructions are possible, they are by no means needed to prove the cancellation property.

In order to prove that, in a commutative ring $(R,+\times)$, the cancellation property holds, you cannot assume that every non-zero element has an inverse; you are not assuming that $(R\setminus\{0\},\times)$ is a group (if it was, your commutative ring would be a field.

For instance, $\mathbb Q[x]$ is an integral domain in checking this means, in particular, that you should check that$$P(x),Q(x)\in\mathbb Q[x]\setminus\{0\}\implies P(x)Q(x)\neq0.$$