A few points here;
1) The most complete listing of the code numbers for dies currently out there is the one I assembled about 10-15 years ago and sent to the forum. It was copied and posted (without attribution, I might add) here ~ http://castpics.net/dpl/index.php/th...die-code-chart
2) The expander die for a 310 Tool should always be separate... it will be one of three lengths, depending on cartridge/case length and these are the KR numbers you see. The important thing is that it have the right length to reach the case mouth and the right diameter. BTW, the special M-type (full sized) Lyman dies are just grown up versions of these dies.
3) NO full length (FL) sizing dies should EVER be used with a tong tool... you are inviting at the least ridiculously hard work and at the worst broken handles. A touch of Imperial™ or other sizing die lube is always a good idea.
4) You should be able to save the die with that stuck case (if the die isn't already scored) by taking the stem out from the top of the die and using a properly fitting punch and hammer to drive the stuck case out. Alternatively, you might be able to drill the primer hole out to accept an Easy Out™ broken screw remover, but that will probably end up allowing the case to score the die if it isn't already.
Yep, I guess you can say I've "been there, done that" and have certainly experienced all of the problems mentioned and more, and have both survived and thrived in reloading!
Regards,
Froggie