If your barrel is not relined, then the proper cylinder would be bored straight through, .375 all the way. If the front of the cylinder is .357, you.ve got the wrong cylinder, that's the one for relined barrell guns. Either way, you should not be getting slit cases with the second reload, I've got .38s that have been reloaded dozens of times before they split.
Who nade your cylinder? You might try posting on the Open tops forum farther down the page, Raven posts there frequently, he's the expert on the conversions.( works with Kirst, making them).
Another thought....Lately Raven and Hoof Hearted have been making some that are chambered for .41 Long Colt case, using a .375 bullet seated INSIDE the case, no need for heel base or hollow base. If you got one of these, then naturally, your .38 cases will split in the oversize chamber. Chamber for that would be about .410 diameter, you could just use a caliper to measure the rear of the chambers to find out if that's what you have.