Usually: the plume (the word comes from Latin for feather) is the name for the volume of matter as it leaves the material. This matter will be in one or more of the 4 states of matter: solid, liquid, gas, or plasma. Typically gas or plasma dominates in the plume. I think "plume vs plasma" processes are to distinguish the fast spray type process from the more general ablation into a cloud/bubble-like thing. Could this be what you are thinking of? https://www.jlab.org/FEL/LPC/05lpc-mao.pdf (slide 3)
... in which case the difference is mostly optical - the plasma process forms a ball while the plume forms a flame-effect.
Usually plasma is much brighter and absorbs the laser light by inverse Bremsstahlung ... so you should be able to confirm it's presence from temperature and/or spectroscopy. The series of slides shows you can tell the two processes by looking at them carefully. You can also see that the characteristic depends on the timescale for the laser pulse used as well as the intensity of the pulse. I'd imagine the crater left would look different too.
Caveat: I am only periferally aware of this stuff so I could have got something wrong.