I suspect they mean "dry" as in "not damp from washing" not "dry" as in "dehydrated". Lots of people wash their mushrooms, which can lead to them cooking down soggy.
You're closer, but I specifically mean to take your fresh mushrooms and throw them into the skillet without any oil (or water). Using dehydrated mushrooms in soups is kinda a shortcut to this.
Though the above comment mentioned Chinese soups and we should recognize that they often use wood ear, which is a different class of mushrooms than most westerners are used to. Wood ear are jellies whereas most western cuisine uses mushrooms with caps and stems (cremini, chanterelles, morels). Still, in general I recommend dry sauteing mushrooms before adding them to the rest of your dish.
More detailed cooking explanation:
Mushrooms are mostly water and air (i.e. a sponge), which is why the traditional French cooking methods tell you to not wash your mushrooms but to clean dry. So recognizing that these are mostly water, really we want to draw the water out, heat (break down cell walls), and then rehydrate with something more flavorful. Most people fuck up on the first step, by using too high of a heat or using something that the mushroom absorbs (oil or water). You start low heat, then the mushrooms release their water, you then bring the heat up to evaporate the water (now cooking the mushrooms). After that, your options change and you need to decide if you want to just brown the mushrooms with oil and seasoning or add them to a sauce or soup. The former will maintain a lot of mushroom flavor but also that nice fat and seasoning. The latter is going to absorb your sauce and add texture and add umami flavor. Either way, you generally reduce the heat here. But I cannot stress enough how mushrooms are sponges and you have to cook under this paradigm. It's like the difference between cooking french toast with bread straight from the loaf or using bread that has been left overnight (or a bit stale). Though maybe this is a bad comparison because a lot of people don't use stale bread for french toast either, despite that being a major part of its history.
I usually look at all the gunk on the mushrooms, and knowing they're grown in a chicken poop mixture, wash them anyway. Probably not better than a good wipe but I can't help myself.
Rinsing them is absolutely fine. The "food science" recommendation around cooking mushrooms has changed in the last few years. Many cooks are seeing good results by starting to sautee the mushrooms in water and only cooking with oil once the water has boiled off. This enables the air pockets to collapse and for a lot of the moisture to evaporate so you end up needing significantly less fats. I've switched to this method myself, and I've been very happy with the results.
Might well depend on the species as to which conditions are most favorable, but AIUI most of your farmed mushrooms are grown on other-than-poop substrates: grain, straw, wood and wood pellets, coffee grounds, shredded coconut husk.
Probably still a good idea to clean them off, though: none of those other things are particularly tasty :)
Most mushrooms you eat aren't coprophilous fungi. You're probably thinking of some variants of magic mushrooms (not all magic mushrooms grow on dung).
The mushrooms you're eating grow from the dirt. If you're into mushrooms, you'll probably also have mushrooms that grow on trees. But unless you're looking to trip, you're not getting poop shrooms.
