The only living part of the nail is the proximal end (the opposite end to the fingertip) and this is where a nail is synthesised and extruded. You can see this clearly if you ever damage your nail and watch over a period of days/weeks as the damaged portion gets closer to your fingertip before growing out.
I've looked into it but can't find any specific term for the portion of the nail plate that extrudes over the fingertip however I'm pretty sure that the reason it is white is because that's the colour of nail. The reason the rest is pink is because the tissue underneath the white, translucent nail is pink. If you've ever seriously damaged a nail to the point of large sections/all of it falling off you can see this clearly.
For further reading on nail anatomy try this reference