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"cut-and-paste" and "copy-and-paste" behave differently.

"copy-and-paste" is as described by Pedro: links to other cells are interpreted as relative unless you specify absolute by using $. So the cell C3 with "=A3B3" when copied to C4 has "=A4B4" which is what we want when generating new content from old.

However if you "cut-and-paste" the cell from C3 to C4 then it retains the links and so will still be "=A3*B3" even though the formula is using relative references. This is what we want when rearranging content. These behaviours are the same whether you cut/copy and paste within or between sheets.

So though you claimed to have "cut and pasted" I suspect you "copy and pasted" - what I'd do as a backup if I didn't realise it made a difference.