I'll concede to semi being the most common term for the whole thang, but I don't believe that it's common to call the whole thang a semi-trailer, at least not for those who drive them. My votes for most common terms would be semi and rig. The semi-trailer itself would probably be called a trailer, the semi part being understood. I would think that anyone calling the semi-trailer by that name would be thought to be a little pedantic but not misunderstood. Owner-operaters are typically only going to own the tractor, whatever they might call it. The trailers are fungible.