The 'Cut or Pluck' debate is old and boring. It does not impact the organism in a meaningful way.

However, if you are collecting unknown specimens for idenitification, its important that you 'pluck' ensuring that the base is kept intact. The base of a mushroom often contains taxonomically significant information.

If you already know the identity of your mushroom, it might be more practical to cut your mushrooms so that dirt and debris doesn't get mixed up with your harvest. Typically I pluck and trim the messy bits off with a knife. Do whatever works for you!

Steps I take when collecting mushroom(s) for identification:

Find mushroom, stand in awe.
Ideally locate multiple specimens of the same species, especially exhibiting multiple stages of growth or types of morphology.
Make note of habitat, surrounding tree species, host or substrate, growth habit (growing in a troop? solitary? scattered?)
Take plenty of clear photos in situ (untouched, as I've found it) and from various angles. I also take photos after the mushroom has been picked, too.
Harvest mushrooms, making sure to excavate the base and keep as intact as possible.
Take a big sniff, note any scent.
Store mushrooms in sturdy basket or other container so that they aren't likely to be damaged on the way home. Ideally keep all species separated (especially from ID'd edibles), I have employed paper lunch bags for this purpose, just make sure you pack them lightly so they dont get squished.
Make sure to not let these sit around when you get home, deal with them right away or you could have a stinky mess if you forget about them (ask me how I know).
Make a spore print while you start the identification process.

Note: Colour is best observed in natural light and before the sample has had any chance to dry as this can affect the appearance. Many pigments and features like warts/scales can wash away in the rain or with age so keep that in mind.

Some handy links:

How To Observe Mushrooms Methodically

An iNat Introduction to Mushrooming

Learn Your Land

First Nature