Both things are using “chain” as the same concept. The sampler or step function defines how to iterate and get from one draw/sample/iteration of the Markov Chain to the next one. From MCMC theory we know that as the number of samples tends to infinite the target distribution will be recovered.
However, in practice we can never have infinite iterations so we never get any guarantee that the samples we have are from the target distribution. What we can do however is identify many situations in which we can be sure they are not. We call this that sampling has not converged. These “convergence diagnostics” generally rely on not generating only a single MCMC chain (a bit redundant but I hope clear) but on generating multiple independent chains. If the samples from each chain don’t represent the same distribution then we know that sampling has not converged (if they do then we continue to not have any guarantee as we still don’t have infinite samples, but it is already very useful to have these diagnostics)
I hope that clarified things a bit, let me know if it doesn’t and if you’d be interested in doing some improvements to the notebook yourself.