You can buy a Moen cartridge puller at Home Depot or many places, cheap.
However, it needs to be threaded into the hole where your broken screw is stuck now.
Like Matt30 said, If you use the plastic tool that comes with a new cartridge, it will let you twist the cartridge left and right, which begins the process of getting it free.
But first, you have to pull out the little horseshoe shaped retainer pin that you can see in all your pics.
It pulls straight up and out.
Looks like yours is already pulled out?
Then you should be able to pull out the cartridge with a pair of pliers, by pulling on the stem.
If still stuck, then you can lock a pair of vice grips (locking pliers), onto the cartridge stem.
Like you show in pic 7.
Then yank hard on the pliers, straight out from wall. That yanking impact should break it out. It might take several attempts.
Push the stem back into the wall, then again yank out hard. And repeat.
If your valve is poorly supported in the wall, this yanking can cause some damage to your tile or the pipe connections.
You could try carefully drilling out the screw, then using the Moen cartridge extractor.
Cartridge Removal Tool for 1200 or 1225 or 1222 Cartridges
https://www.homedepot.com/p/202998692
You can find lots of videos of “how to remove stuck Moen cartridge” on Youtube.