Although the three terms do have different connotations, for "routine" magnetic resonance imaging (MRI) they can be considered largely synonymous.
Routine clinical MRI records the signal from hydrogen (¹H) nuclei, principally those in water and fat molecules. And since the ¹H nucleus consists of just a single proton, we sometimes say the MRI signal comes from protons.
Spin angular momentum is an intrinsic quantum property possessed by individual protons and whole nuclei allowing magnetic interactions and resonance to occur. So often the protons or nuclei possessing spin and giving rise to the MR signal are simply called spins.
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To make matters even more confusing, sometimes the word spin refers to the magnetic properties of a group of similar protons rather than those of a single proton. Here, the word spin is an abridged form of a longer term, "spin isochromat". A spin isochromat (literally "spins of the same color") is a concept that allows us to talk about groups of individual spins that behave in a similar fashion. This notation is convenient because it frees us from constantly coping with the peculiarities of quantum mechanics in explaining the more general aspects of NMR and MRI. Specifically, we don't have to deal with the quantum uncertainties of a single proton, such as knowing exactly which state it is in or in precisely which direction its magnetic moment is "pointing" at a certain time. Since the spin isochromat represents not one (but many identical) protons, its behavior can be considered equivalent to a quantum average or expectation. The spin isochromat can therefore be treated as a macroscopic magnetization that can be analyzed according to the laws of classical electromagnetism rather than by those of quantum mechanics.
Hendrick JE (ed). Glossary of MR Terms, 5th ed. Reston, VA: American College of Radiology, 2005.
What is spin?
What is the difference between "spin" and "spin state"?