I'd bet that you need the official tooling, as recommended by the genuine connector manufacturer, to get proper bellmouths at both ends of the conductor crimp. Those of us with shallower pockets using cheaper tools are doing well to get a conductor crimp that's uniform for its whole length, and without excessive distortion. Cheap sets of dies made by wire EDM cutting consisting of pinned together sections are almost invariably only going to give you straight crimps.
On the terminology; Why am I not surprised that sections of the electronics industry have done their own thing, the opposite way round to the rest of the metal forming industry? @Tooki: Thank you for making me better informed.
As to wire size - if the tool or its instructions, or the contacts datasheet give a size range, stick to it, however for cheap tools that at best give a vaguely marked size on the jaws, its 'suck it and see', and check it passes a pull test. If you are buying wire from a reputable supplier, it should meet its specs, including cross section area (CSA), but from dubious suppliers, you'll often get less copper than you expect, and sometimes far less! Salvaged wire is down to you, does it look right compared to the good stuff, and if its critical, one may have to break out the micrometer, measure enough strands to get an average, count the strands and calculate the total CSA to be certain.