This all up based on "common sense," and inference, I have no fucking idea and I have not looked into this - the main problems associated with cutting as a method of CTB, to my knowledge are the fact that it can be extremely painful, and that your blood is highly likely to clot before anything happens - if the pain doesn't rule it out as a successful attempt, then you are extremely likely to fail anyway.
There seems to be a misconception that being in still water will decrease blood clotting. But apparently the clotting process performs perfectly well in the presence of water:
"Blood clotting is an active, biological response that is triggered by damage to the wall of any of our blood vessels. First the muscles around the vessel contract to slow the flow of blood there. Next tiny blood cells called platelets attach themselves to the injury site and begin clumping together. This clump is reinforced with strands of a structural protein called fibrin. Other proteins crosslink the fibrin strands to harden the gelatinous plug. All of these processes will proceed perfectly well underwater but because water is dense and a good solvent, fast flowing water might wash away the clotting factors before a stable clot can form on an external cut."
So in other words, water isn't really going to help you. Running water (which would probably increase localized discomfort around the cut(s)), however, can help slow down superficial blood clotting towards the surface of the skin, however, the deeper areas of the wound (which is the part that probably matters more), will clot regardless because the clotting factors wont be disrupted beyond the surface of the cut. In general, being warm will increase blood flow and dilate blood vessels, so keep that in mind.
Common sense tells me you should learn about anatomy, and shoot for arteries, in fact, I'd imagine it is absolutely crucial. Cutting the same artery in several locations seems like it would decrease the chances of successful clotting. But obviously, this method is based entirely on blood loss or something caused by significant blood loss (as I can't think of any other modality in which lethality would result from cutting). You need to lose a fuckton of blood to die - so I imagine a lot of cutting related deaths in baths probably are actually caused by drowning after passing out from blood loss or sedatives, not actually from blood loss.
Any obvious "technique" that would increase blood loss would probably be a "good" idea: using a very sharp instrument, using a larger instrument, cutting deeper, cutting longer and further... but all of this may increase your SI significantly anyway, or rule it out all together as something you think you are capable of.
I would imagine that taking a blood thinner, or even better, an anticoagulant could help, but is this effective enough?? No idea, my guess is no. If you wanted to get these drugs, how would you get them? No idea. If you did take these drugs, how much should you take? No idea.
A major issue associated with cutting as a method of CTB is pain. I know some people think they can handle this, and it may even attract them to the idea if they enjoy cutting or SH in general, but cutting with the goal to CTB will need to be significantly deeper than most people are used to, and this will increase the chances of hitting a nerve I'd imagine. That is not the type of pain people who SH usually are looking for - it is a sharp, burning, and extremely acute pain that I can't imagine anyone tolerating (as a IV drug user that has accidently hit nerves on rare occasions, it fucking hurts, and it hurts in the weird unique, fucked up way). Do I understand anatomy enough to tell you that cutting a nerve is even a serious concern? NO. I guess taking pain killers of any kind would increase chances of success. However, I'd imagine if there's like a 3% chance of success usually (which I just made up for the example), I'd imagine doing all of these things would still keep the success rates extremely low.
Since this method involves such a low success rate, and a failed attempt may result in a serious, obvious open wound that may need stiches or other medication, survival would almost definitely risk serious infection and scarring. If you seek the medical attention you need, you risk indefinite institutionalization in some facility for mentally ill people. In fact, I'd absolutely count on getting admitted.
I am also curious about this too, but it is a method that is highly likely to bring you pain, both mental and physical, and a ticket to the psych ward.