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.

How to take a spore print:

I like to use aluminum foil to print on, it tends to show light and dark coloured prints. I cut a long rectangle so that I can fold it up and store in my spore collection. If you use white paper you might have trouble seeing white or light coloured spores, same goes for dark paper and dark spores. I have heard of people using clear glass too. It's really up to you.
Cut the stipe off of the mushroom you'd like to print. If theres no stipe, you can leave as is but sometimes with pleurotoids I will break the cap so that it lays flat. The goal here is to have the hymenium flat against the foil.
Place the cap, spore surface down, on the foil/paper/glass and cover the cap with a bowl to stop any drafts from disrupting the flow of spores onto the surface.
If the mushroom is starting to dry out you can add a drop or two of water to the cap surface to increase humidity which aids in spore release.
Wait 12-18hrs and check, I have left prints going for 48hrs if the spore load isn't very heavy. Some species don't drop a whole lot of spores and some drop a LOT.
If you want to save the print for later reference or microscopy then allow any condensation on the print to evaporate before folding up the foil. Label with date, location found, species and the collector's name before putting a plastic zip bag.

Note: I have found that too much humidity can sometimes be detrimental, especially with brackets and conks. This usually presents as a very faint or nonexistent spore print and a lot of condensation on the foil. In these cases I like to print these without the bowl on top. Sometimes I will drape a paper towel over top of the cap. Just something to keep in mind. You will find your own little methods and tricks to get great spore prints.

Some handy links:

How To Observe Mushrooms Methodically

An iNat Introduction to Mushrooming

Learn Your Land

First Nature