If you are working from an older field guide, every inkcap in it is called Coprinus. That has not been correct for over twenty years, and if you use the old names when asking for help online you will get confused answers.
What happened
Molecular work published in 2001 showed that the old genus Coprinus was not a natural group at all. The deliquescing habit - the way the gills dissolve into black ink to disperse spores - had evolved more than once. The genus was split.
Coprinus itself was kept for a small group centred on the Shaggy Inkcap, Coprinus comatus. So the Lawyer's Wig is one of the very few that did not change name.
Coprinopsis took the Common Inkcap, now Coprinopsis atramentaria, along with the Magpie Inkcap and many of the larger grey species.
Coprinellus took the Glistening Inkcap, Coprinellus micaceus, and its relatives.
Parasola took the delicate pleated ones that appear on lawns overnight and are gone by lunchtime, such as the Pleated Inkcap, Parasola plicatilis.
Why it matters in the field
The Common Inkcap, Coprinopsis atramentaria, contains coprine. On its own it is edible, but coprine blocks the enzyme that breaks down alcohol, and if you drink within a few days of eating it you get flushing, palpitations, nausea and a pounding headache. The reaction can recur with a later drink. That is a good enough reason to leave it alone entirely.
The Shaggy Inkcap, Coprinus comatus, has no such problem, but it must be picked young and cooked within hours. Once the cap edge starts to go pink and then black, the whole thing turns to ink in the basket.
The awkward one
Galerina marginata, the Funeral Bell, fruits on wood in small clustered groups and has been mistaken for young inkcaps and for other small brown mushrooms. It contains the same amatoxins as the Death Cap. Any small brown mushroom growing on wood deserves the same suspicion you give an Amanita.