In the 357 Bobcat , the cylinder in that conversion is chambered out to a full 44Magnum dimension. Then the small plastic sleeves are placed onto the individual bottlenecked 357-44Mag necked down cases to make them take up the space of that 44Magnum chamber.
The bbl is 357cal of course.

The 357 Bobcat will accept a 44Mag/44Spl round in the cyl as it is,,and you have a 357cal bbl on the gun in front of it.
No one doing a conversion to the caliber necessarily follows up with a caliber designation change on the bbl or frame.
Got to pay attention..

The Bain & Davis version has the chamber cut to the bottleneck dimension of the reformed case. So a 44Mag or 44Sp case will not fit into the B&D chamber.

The idea of using the plastic sleeve on each cartridge in the Bobcat version was to avoid them setting back and jamming the action up.

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I just stick with a .357Magnum,,or a 44spcl myself and the problems end right there. But for those who have the need for speed,,this was one way to that end.