There was a competitive application process to get into this Americorps program and I did receive a living stipend (barely enough to cover basic living expenses). I'm inclined to list my Americorps time as professional for that reason but I was wondering if there's a good reason not to.
I wanted to split this question up into two parts (resume and LinkedIn profile) because a resume and LinkedIn profile seem to serve slightly different ends and may thus have slightly different answers.
On a resume you have the flexibility to not make artificial distinctions that don't serve your needs. Instead of "professional experience" and "volunteer experience", you can just list "experience". I've seen resumes that list internships, time in the Peace Corps, and mandatory military service under "experience". If it's relevant experience or if it fills a resume gap, then listing it benefits you.
I recommend making it clear that it was a volunteer position; you didn't work for Americorps Inc, after all. If your role there had a title then you can list it as "Assistant Data Wrangler (Volunteer), Americorps". If it doesn't have a title then you can list it as "Volunteer, Americorps". (If the position did provide professionally-relevant experience, though, like an internship does, I recommend finding a way to add a couple words to this line about what you actually did.)
The job of the resume is to (accurately and positively) describe your qualities and background so that you'll get the interview. Any format that does that will suffice. As one example, I'm currently listing my position as a Stack Exchange moderator (because it involves skills that are relevant for my job search), but because it is a part-time volunteer activity I list it at the end of the experience section, out of chronological order. So it's there but not more prominent than it should be. (I list it using the "title (Volunteer)" format I recommended.)
As for LinkedIn, I haven't seen a lot of consistency in the volunteer section. Some people list full-time unpaid activities in that section; others list those in the experience section and use the volunteer section for smaller-scale commitments (things that you might spend a few hours a week on). List it where it seems to fit for you.
Finally, I don't think how much you were or weren't paid is the key factor here. The skills you learned and used, and the amount of time you spent on it (particularly if full-time), seem much more important.
