For some types of positions like this, Silver, we use the word "internship" to talk about unpaid or poorly paid jobs that young people might take with a company in an effort to gain experience in an industry. She had an internship with the Mobile Corporation this summer.
I'd probably use something like "I have a volunteer position (not volunteering) with the X organization." This could refer to any unpaid function that I was performing for that organization. Or "I volunteer with/for X." This version is probably the most common.