What is the definition of a cross stitch sampler?

The "sampler" usually refers to your finished product. Some samplers are small and very simple. Others are larger, and very ornate. But the term is usually used to refer to whatever it is you are working on. The term Cross stitch sampler started hundreds of years ago as young girls in school were learning alphabets and spelling and would use linen to spell out these Words. Some added their own design with thread such as monograms, motifs and other touches.

Band samplers are rows of stitches (bands) usually of different specialty stitches such as Algerian Eye, Rhodes, Smyrna etc.