The idea of the hot water is to heat up the barrels so that they dry out quickly and don't leave moisture in the nooks and crannies. Cold water will dissolve the residues perfectly well. If you use cold water then you need to dry it very thoroughly. I’ve never heard of wax finish on the barrels - on the woodwork yes, but not the barrels. Is your gun an old one with brown Damascus barrels or a latter one with a blacked finish. I only wipe the metal work over with a slightly oily cloth. The browning or black/blueing finish is to prevent it rusting and only requires a wipe over with an oily cloth when dry.
Looking back over what I suggested for cleaning I didn't mention that you should always remove the barrels from the woodwork before cleaning - usually there is a peg holding the barrels on to the for-end wood and a hook on the rear of the barrels - just push out the peg - not all the way as they are usually retained -and then lift them off the hook.
