Slow cooked steak

As you mention, cooking the steak to medium rare does indeed kill the bacteria on the surface of the steak which is where most of the risk of contamination exists, so immediately after you've cooked the steak, if you eat it, you're probably pretty safe unless there happened to be bad organisms living inside the meat.

Assuming that you've got a good piece of meat, with nothing terrible inside it, after you cook the meat, what you need to worry about most is bacteria that you introduce to the meat, and so to answer your question, cutting meat shouldn't affect the safety of storing meat after cooking at all (as long as you use a clean knife/cutting board).

While cooking the meat does initially kill all of the surface bacteria, the fact that the surface was once cooked doesn't really do anything to deter new bacteria from moving in and going to town on the meat. What this means is that you need to be very careful to keep meat in sanitary conditions regardless of whether the exposed surface has been previously cooked or not if you're planning to eat it without cooking it again.

So, to recap, if you take a piece of meat where all bacteria has been killed via cooking, and seal it without introducing new bacteria, you should be safe to eat it.

The one remaining issue is that we're not talking about a steak that's necessarily had all bacteria killed. There's a possibility that some bacteria survived the cooking inside the meat. If this is the case, you may be safe to eat the meat right away if there are very few of these bacteria present (depending on what they are), but the longer you wait - even if you seal the whole thing up - the longer you're giving those bacteria to multiply into large enough quantities that they can destroy the meat and/or sicken you. To avoid this, you want to put the meat in the refrigerator, as soon as possible to slow bacterial action, and eat it as soon as possible. Like John, I've had success with around 3-5 days, but it really depends on what you're starting with, so I'd highly recommend that you have a good look/sniff before you eat to see if there are any signs of spoilage and discard if so. Again, this is a risk whether you cut the meat or not, so that isn't really a factor here.

Regarding partially eaten steaks, this is just introducing one more place where bacteria can get on the steak before it's sealed up and refrigerated. I could see this being no issue (if you're cutting off a piece of the steak, and sealing/refrigerating the rest while eating), or a significant issue if you're cutting the steak with utensils that have been in your mouth, or worse, trying to store a piece of steak that's been in your mouth. In those cases you're almost certainly introducing more bacteria and decreasing the amount of time you'll have before spoilage occurs.

So, in summary, cut the meat if you like - that should make no difference. Get it sealed up as soon as possible without exposing it to anything non-sterile. Put it in the refrigerator as soon as possible. Throw it out if at any point you detect spoilage. Following these steps you should be safe to keep your leftovers for 3-5 days and only rarely need to discard spoiled food.

If there really is fat inside the steak, it is terribly undercooked. But this is unlikely to happen.

Animal fat is a saturated fat which melts somewhere around 35°C. If your steak hasn't reached 35°C internal temperature, it will be very visibly raw on the inside, with no apparent difference from the cold steak from the fridge. It will still be oozing blood-like liquid even after resting for over 10 minutes, and the liquid will be thicker than the juices of cooked steak. It will be deep red and slimy, and cold to the touch. Even the rarest steak you can buy is taken above 40°C.

But not everything white inside meat is fat. Connective tissue/fascia are white too. They are not really lump-shaped, but I don't know how precise your description is. Try cutting through one of the lumps. If it is tough and rubberlike, it is connective tissue (and if there is lots of it, your meat is not meant for roasting, but for a slow cooking method). If the knife goes easier through the lump than through the meat, it is fat, and your cooking method is seriously wrong.

In any case, the best way to know when steak is done is to use a meat thermometer. Stop when it shows around 61°C, the remaining heat transfer after removing from the heat will give you a medium steak.