The no-cloning theorem itself can be stated very precisely.
Given an unknown pure state $|\psi\rangle$ that is drawn from a distribution
$\{p_i,|\phi_i\rangle\}$ (known to the counterfeiter), it is impossible to create a perfect clone with unit probability
unless all the states are orthogonal,
$\langle\phi_i|\phi_j\rangle=\delta_{ij}\ \forall\ i,j$.
Of course, as you say, there are strategies that either (i) succeed perfectly but with some probability of success less than 1, or (ii) always succeed but create the target state with some finite accuracy, as measured by the fidelity (F<1). In my experience, (ii) is the more common case.
What these parameters are is highly dependant upon the distribution that the states are drawn from, and there's no universal answer. The sort of cases you will find that have been explicitly calculated include:
- Universal cloning, where the distribution is uniform over all possible pure states of a fixed Hilbert space dimension.
- Equatorial cloning (this is my name for it, many in the literature use 'phase covariant cloning' but I find that misleading). The system to be cloned is a qubit, equally likely to be any pure state on a great circle (usually the equator) of the Bloch sphere, i.e. $(|0\rangle+e^{i\phi}|1\rangle)/\sqrt{2}$ for any $\phi\in[0,2\pi)$.
- Phase covariant cloning (again, my name for it). A qubit state $\cos\theta|0\rangle+\sin\theta e^{i\phi}|1\rangle$ for some arbitrary distribution $f(\theta)$, but still uniform over $\phi$. In other words, the distribution is invariant under rotations around the z axis of the Bloch sphere.
- A discrete pair of non-orthogonal states.
You can derive these success parameters for arbitrary numbers of clones, and you can even demand that different clones have different fidelities, and derive the optimal trade-off between them (full disclosure: I've worked on that quite a bit).
I should probably emphasise that what you want from cloning is highly dependant upon application. In a lot of theory, it is the simple fact that perfect cloning is impossible that is useful within other proofs. Figures of merit are only really relevant if you have a concrete scenario where you want to implement approximate cloning. I'm not aware of many. Note that, for example, it's not so useful in security proofs of e.g. quantum key distribution, because that might allow you to bound how well a particular class of strategies perform, but how do you know there aren't other strategies not included in your analysis?