"re-profiling the edge"
"setting the edge"
"setting the bevel"
These three could be used interchangeably. Basically, it just means regrinding new bevels to establish a new, crisp edge shape, a.k.a. 'setting the geometry' of the edge. This would be done in the case of repairing a badly damaged edge, or changing the edge angle to suit your preferences.
"sharpening"
"honing"
These two could also be used interchangeably. 'Honing' means 'sharpening', and 'sharpening' means 'honing', for all intents & purposes. Most often, these would apply to the regular maintenance of an edge, without necessarily doing any major regrinding, as with reprofiling mentioned earlier. Often done at medium or finer grit levels, just enough to make it functionally sharp again.
"stropping"
'Stropping' technically refers to using a leather 'strap' with an edge-trailing stroke (opposite of the edge-leading stroke done during 'sharpening' on stones), to remove burrs or very gently hone or polish an edge; usually done with very fine abrasives on leather, or on bare leather alone. Could also be done using fabric 'strops' with or without compound (canvas, denim, linen), or the term is sometimes applied to 'backhoning' on wood or paper, usually with very fine compounds.
"refining the edge"
This could apply to anything done to make the edge even thinner, sharper or more polished, or all of these 'refinements' in combination. Could be done using stones or strops, usually at finer abrasive levels and with a very, very light touch.
David