The best way is to look at the back side. Leather will be very obvious and look like a suede. Vinyl will have a woven or fuzzy fabric background. If you cannot get to the back, you can try the pinch test. Pinch some of the material between your fingers and rub the backsides together. If you have discriminating feel, the vinyl will be more slippery than the leather (if you can imagine rubbing the above described two surfaces together).

From the outside it's difficult to tell because leather can be corrected, grained, and top coated with an acrylic color coat.

There is also a product called re-bonded leather. The best way to describe this is re-bonded leather is to hides as particle board is to lumber.

And you are right a "leather-matched" piece has leather where you touch it and vinyl in places where you don't -- outside arms and outside back, sometimes front panels.

Vinyl will have a certain amount of suppleness. A really hard material is probably an inexpensive leather. Leather will also smell like leather, vinyl won't.