............I picked up on the threaded collar and figured it had to be an "L" type. My 11" Logan has an L00, but I'd suspect your big ole brute has an L0, or a L1. Not any lathes made these days using the L nose that I'm aware of. They've all gone pretty much to the 'Stud' type D camloc on the lathes of any size, with the smaller imports mostly having threaded noses. . Maybe a couple using the short taper A types, but I haven't heard of who if any makes'em.
I grind those form tools or "Spoons" for forming my swage dies. Since there are no lube grooves I use HSS steel. One of these days I'll have to try making a grease groove slugYou're using plain high carbon steel?
The tool in the previous photo was for making 58 cal swage dies, but the same type was used for this slug for the 38-55.
You might be interested in trying half drills ("D" reamers). They're HSS, already ground in halfThe above are 23/64 (.359") and would be good for .375" and larger. Of course you cut off the excess as they're too long. They're ferociously expensive to buy retail but you can find them on E-Bay from time to time pretty reasonable. I was able to purchase 2 packages (12 drills) of these for $29 + shipping. MSC currently lists this size (Made in USA) for $32 each. There are currently 4 different sizes on E-Bay.
.................Buckshot