I can't believe I remember all this stuff from working at Moen a decade ago, its still in my blood all these years later. Granted email and message boards stink when discussing technical things, visual is always better, but alas, here ya go.
You're seeing the plastic cartridge nut in that last photo, that's the white plastic outer diameter ring with the two cut out tabs. The D-shaped stem with the hole in it is the cartridge stem, there is an outer cartridge body there too, but it's hidden under the white plastic cartridge nut. Basically the white plastic cartridge nut screws down into brass valve body to the point where it hits the top of the cartridge to hold it in place. Without that nut, the cartridge would blast out under water pressure. You'll note on the top of that nut there are two cut out tabs directly across from each other. There is a tool that Moen has that mates with those two tabs and fits like two puzzle pieces going together and from there you can rotate the tool and the nut screws out. Then the cartridge can be pulled straight out with a needle nose pliers. Getting back to the nut since you don't have the tool, now I've heard people getting these nuts out by hand when they're loose, just by unscrewing them out, but if there in their tight, it ain't easy doing it by hand. I've also heard people using a flat bladed screwdriver to angle into one of the tab indents of the white plastic nut and tap it to loosen it, but again, you don't want to damage those indent tabs because the job will be even harder if you do. The tool makes it a heck of a lot easier. Of course I'm stating the obvious here, but anytime you're dealing with a roman tub, make sure you have the water off first before playing with this stuff.
The cartridge part number is either a 1248 or 1248B (they are the same cartridge, the "b" stands for bag, meaning it comes in a bag) or you need cartridge number 14891 which is probably more likely given the age of your product. The 1248 came out in the 90s I believe so your product is 1960s/70s era Moen meaning the 14891 is more likely. If the problem is a drip after shutting the faucet off, the cartridge will fix it. If you have a leak under the handle while running it, the cartridge once again is the fix. Hope this helps.
Forgot to add, when reinstalling the cartridge, it only goes in one way, sure you can slide it all the way in, but you have to be mindful of the locator tab, see drawing below where the arrow is. The locator tab has a corresponding cutout in the brass valve body so when the tab mates into the valve body cutout its properly seated all the way down. Sure you could just plop it down in there and screw the cartridge nut back down, but you run the risk of handle leaks and the ports not aligning properly. The trick typically is to push the cartridge down in there just a hair off center of where the cartridge port aligns with the valve body outlet port and then rotate the outer cartridge body (not the D-shaped stem of the cartridge) a hair and you can feel the cartridge drop slightly then you'll know your in the groove so to speak and it's fully seated, then reassemble and you're home free. When you rotate that cartridge to seat it, you're not rotating the white plastic D-shaped stem, that rotates separately from the outer body of the cartridge. Sometimes when putting the cartridge back in you have to screw in a stem extension ( of which I don't see in your photos) on top of the cartridge to help you rotate it to get into the tab. The cartridge seats so deeply into that valve you can't exactly put your fingers down there to rotate the cartridge body so having a stem extension on of the cartridge helps in the rotation of the cartridge body once you push it down deep into the valve. There have been cases, where customers do there best to line it up and push the cartridge down straight and it aligns right of the bat without the need for rotating or stem extensions so that might be your best option of just trail and error for seating it. You'll know if you succeeded because, as I mentioned earlier, it will leak especially around the handle area.