While this answer might seem long it is unfortunately not as in depth as I want. I have not cut any ceramics with the tools and techniques I am going to suggest. Given the desired end result I feel that this would be an approach to consider though.
This might be difficult depending on your level of expected precision and tool availability
I have a suggestion but it involves tools that could be expensive to acquire and potentially a difficult process (I have thoughts on that as well near the bottom). The cuts you propose and the shape of your mugs will affect this as well.
Consider wet tools
When it comes to cutting materials like glass or ceramics with power tools, wet tools are always advised. You don't want these material fragments stuck to your cutting instruments. Nor do you want the inside of the machine to be exposed to them. Aside from hand tools, that would not have this issue, I would suggest that any power tool you consider be a wet tool.
Easy tools to comes by for this type of task would be a wet tile saw or a ring saw. Both are for cutting ceramics and glass (as well as some other materials). However neither common variety will have the thickness clearance of your average mug.
Therefore a wet band-saw would be a good suggestion here. While there are not too many of these available from what I could tell the real issue is trying to get precision cuts with one of these devices. Of the 4 common ones that showed up in my research they had mostly plastic frames (Likely due to the use of water) and only a couple had what I might consider a mitre slot (Which would be useful for making custom bandsaw sleds). This is important part of the precision I think you would need.
Note: The above picture is not a wet bandsaw. It is used to highlight the presence of a mitre slot.
Consider the following related video where a bottle is cut with a wet bandsaw
The problem with mugs is, depending on your cuts, you cannot make them perfectly straight free hand. At least not without practice experience. This is a major factor here since you intend to reassemble them after. This is also a hard topic to expand on as every mug and every style of cut might need a specialized jig.
Consider the features of this jig\sled on a bandsaw.
The wood in the picture is being held against two fences. Rear fence holds the work 90 degrees to the blade and the other fence is adjustable (see the wheel) to help move the cut in or away from the blade.
Your cups are not flat (at least not the one pictured) so you will need to modify this design. When cutting into curves you need to prevent the cup from rolling around so an easy way to get that is to have two blocks angled into each other (like chucks on a car or plane wheel).
Made in sketchup to illustrate the design
Cup has somewhere to rest and the adjustable fence will give you something to push against to make sure the cut is straight. That design only covers your first cut examples though.
Getting mugs of a uniform width makes this part so much easier to mitigate. A jig would be useful if you were using power tools or not. The idea is to place the cut in a certain position where it will not move unpredictably while you are cutting it. If you are doing it with hand tools then clamps would help as well. Clamps would be good for power tools as well but their position could be awkward and you don't want to risk damage.
Regardless of your cutting tool, powered or not, you are going to have an issue with the kerf(width of the cut) of blade. Let say that you cut the mug down the exact middle of two identical mugs. Lets assume that your blade kerf is 1/8". When you go to put that mug together it will now be ~1/8" smaller so the edges of the mugs won't line up.
Not the end of the world obviously. You can address this but cutting off center on your mugs. On one mug you could cut a kerfs width to the right of the cut line and the other a kerfs width to the left. This also means you could have to cuts extra mugs to get the pieces you need. This also assumes your level of perfection your are trying to achieve.
When you are cutting make sure you cut slow. Make sure you cut slow as to not force the blade askew or cause any burning.
I think it is doable with power tools but unless I am way off the mark there would be some work involved in getting the right environment to do this properly . If you are willing to concede on some degree of perfection and willing to practice I think hand tools might be the way to go like Chris H suggests. Several caveats above would still apply. You might, in either case, consider food safe grout to make up for any size consistencies.
It is still feasible though how I describe.
Access to tools/workspace
There are locations that are set up for you to have access to industrial tools. So if purchasing them outright is not in the budget or you are just curious to test; look for something like this in your area.
TechShop is one of these public workshops that, with a membership, gives you access to tools and work space. They have locations all across the United States (which is obviously not useful for all). There are other places like this that exist for the purpose of tool rental and tutorials beyond what you would get from something like a big box store. TechShop is just an example.