I can use the screen as a viewfinder or the smaller, eye viewfinder on most cameras. Is there a proper name for the eye viewfinder?
It's simply called the viewfinder. It can be optical (on an SLR) or electronic (on a mirrorless camera, basically it's a second, smaller screen). Don't say viewfinder when talking about the back display/screen, that will only confuse people - the terminology for using the display to compose your shots is live preview.
Typically viewfinder means the one you bring to your eye, and the screen on the back is more commonly referred to as the screen or LCD. :)
If you want to be more specific, you can call it an eye-level viewfinder to make it distinct from using a flip-up LCD as a waist-level viewfinder, like the ground glass screens on TLRs.