Here is an attempt to give a summarized view of all the interesting perspectives that have been kindly shared here.
People vs. content :
When promising or offering anonymity, the focus can be put on the person ("You will be anonymous") or on the content ("Your posts will not reveal your identity"). This already makes a difference as the latter is less likely to convey the idea of being cloaked. Phrases like "leave your post unsigned" can be used for this purpose.
Choose what you prefer :
Being offered anonymity as an option on a post-by-post basis implicitely means that posting under one's real identity can sometimes be fine or even desirable, which keeps the focus away from intrigue and stealth.
Cloak and hide vs. be generic vs. use a cover name vs. speak for the group :
At least four different types of promises emerge, each with a different psychological impact.
- cloak and hide : the promise is basically "No one will ever know who you are", which conveys the negative connotation of intrigue and stealth. Phrases like "Remain anonymous, incognito" tend to go this way. More neutral alternatives can be found though, like "unidentified" or "private".
- be generic : the promise is "You will go undistinguished and appear as anyone doing the same", which starts shifting the focus from "You can do bad things without any risk" to "You will be generic and thus will not get personal credit from what you share". Phrases like "Sign post as a guest, a fellow contributor" carry this promise.
- use a cover name : the promise is "You may use a cover identity that will allow you to appear as a person but will act as a shield". Cover identifiers like "user726, contributor726, visitor726" provide an anchor for the readers of the post to "identify" the anonymous writer.
- speak for the group" : the promise is "You are part of the group and may speak in its name", which means that the user will blend into the group but be valued and respected enough to be able to speak for the group. Phrases like "Vox populi, speak for the group, send community post" go this way.