If I expect a maximum central venous pressure of 5 mmHg in a peripheral vein, why doesn't water enter the system if a diver underwater injures themself?
You're confusing relative and absolute pressures, which is easy to do because we don't usually specify which is meant, the meaning is implied by context.
At sea level, atmospheric pressure is around 760 mmHg. If you take "normal" arterial blood pressure to be around 120/80 mmHg and venous pressure to be closer to 5 mmHg, that would suggest an extreme outward pressure on both arteries and veins, where any cut would result in a massive rush of atmospheric gas into the vasculature. Of course that doesn't really happen. Why?
When you say the venous pressure is 5 mmHg, you mean relative to the outside. Relative to outer space, the pressure of blood is more like 880/840 mmHg in arteries and maybe 765 in the veins. Whether the pressure outside is from atmosphere or atmosphere+water, the pressure described is relative to that outside compression.
If you go underwater, pressure increases on the outside, but that pressure is also pushing on and compressing all the vessels in the body. There's nowhere for the fluid inside to go, so its pressure relative to outer space increases, but pressure relative to the local outside won't really be any different. Just like in atmosphere, most of the absolute pressure in the veins is not from any sort of muscle tension or compression within the venous system, only the 5 mmHg comes from that - the other 760 mmHg comes from all the weight of the atmosphere, and if you somehow opened a magic portal from the inside of your veins to outer space, yes, in that instantaneous moment blood would come out with a lot of force, and it would come out with even more force if you were deep under water.
Diving pressure is important for the dissolved gases in blood, though, which are compressible unlike the liquid blood.