in my opinion "voluntary work" = "volunteer work" and I think the first one is mostly used in BrE and the second one in AmE, but I`m not sure about it.
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.