The following may have been the 'other explanation...
I have no idea what the evidence for this is, but there is a suggestion that, in America, the figures to be written on a cheque (or check, as they would call it!) were sometimes written so heavily that they, in effect, �tore' the paper. This was supposedly a way of ensuring that they could not be altered.
Another suggestion is that some cheque-books had several cheques printed on single pages. These had to be torn along the perforations in order to remove individual ones. (I have personally never seen such a cheque-book, but they could conceivably exist!)
I still think cut = prepare is the answer.