It looks like a B to me. I have a CK and a C and that one has aspects of both - but it's hard to tell from one picture. You aren't going to enjoy changing that ring gear. To get it on the trailer, you can try the Delmer approved method of putting it in neutral - making sure it is not able to start by pulling the fuel shut off even though it is unlikely to start by rotating it so slowly - and reach in and grab a fan blade and the fan belt at the same time and rotate the engine to a spot where the ring gear is still good - and give it a go. Don't forget to push the fuel shutoff back in before cranking it over. You may want to consider more of his advice here: https://www.heavyequipmentforums.com/threads/case-580b-flywheel-question.38789/
I recently bought a new starter for my CK. It has the 188D diesel engine. I don't have the number handy but I could come up with it in a day or so if necessary. I found a couple different types of starters - one was like the original starter and the other is a high torque / gear reduction type. I got the second one. It's supposed to be a better design. I also read somewhere that someone's 10 tooth could replace the stock nine tooth because the shaft was offset. Don't ask me how that works but it probably does.
Your starter may be fully engaging but then running into an area of damaged ring gear so a new starter may not help much and changing that ring gear is a big job. I'd change the whole torque converter, rebuild the shuttle, change the clutch, etc. if it has one, and check the ring and pinion while the tractor is split. You won't want to go back in anytime soon. Good luck.
I recently bought a new starter for my CK. It has the 188D diesel engine. I don't have the number handy but I could come up with it in a day or so if necessary. I found a couple different types of starters - one was like the original starter and the other is a high torque / gear reduction type. I got the second one. It's supposed to be a better design. I also read somewhere that someone's 10 tooth could replace the stock nine tooth because the shaft was offset. Don't ask me how that works but it probably does.
Your starter may be fully engaging but then running into an area of damaged ring gear so a new starter may not help much and changing that ring gear is a big job. I'd change the whole torque converter, rebuild the shuttle, change the clutch, etc. if it has one, and check the ring and pinion while the tractor is split. You won't want to go back in anytime soon. Good luck.