That there are several articles about people reverse engineering the client and protocol and nobody else has brought this up, I feel pretty comfortable calling this unfounded unless you can provide some indication otherwise.
Given that that information does go to the server, there's no way to know what it's used for.
That "given" there is doing a lot of work. I've never seen any reference stating that it's sending the entire process list anywhere.
The current third party protocol implementations all expose setting the game/activity you're playing directly, sure making it look like everything is done client-side.
You would have to show all network activity from the app never includes this type of information or anything derived from it, because as soon as anything goes server-side it can be used for anything. Is this actually easy?
All any of the people claiming that discord is spying on you need to do is capture one packet going out that does have that sort of information, or find one piece of code in the client that supports doing so. That is the part that I think would be relatively easy given that it's primarily just a gob of javascript in a chromeless browser.
But that's the point. There is a "server" running on your machine that is snooping on your processes (therefore the one-click integrations). The "client" you interact with never sees this info, instead just what the "server" exposes to it, so I claim you'd need to reverse engineer the "server" or wireshark your network and read probably-encrypted communications...