The traditional language of MCMC which talks about `algorithms' and `samplers' constrains inventiveness when taken seriously. It is much better to talk about `updates' or `update mechanisms'. The point is that updates can be combined in many ways to make an MCMC algorithm. But talking about the `Gibbs sampler', for example, suggests focus on an MCMC algorithm composed exclusively of Gibbs updates. Nowadays, the only reason for that is sheer ignorance. Gibbs updates are often `part of the problem rather than part of the solution' as we used to say back in the '60s. I have often found when asked for help in improving an MCMC algorithm that my suggestion was `Why are you using that Gibbs update? That's what is making the sampler so slow.' As soon as the questioner thought about using something other than a Gibbs update the problem became easy. Another quip in the same vein is Peter Clifford's characterization of the Gibbs sampler as a cul-de-sac that many statisticians are now trying to reverse out of (Clifford, 1993).
Even when the terminology of algorithms rather than updates does not lead to construction of inferior Markov chain samplers, the terminology still makes description of algorithms more complicated than it need be and makes it harder to talk about Markov chain samplers. For these reasons we always describe the `basic update mechanisms' that are combined to make the sampler.