No, Nvidia is not providing any driver for the intel gpu. Maybe read this:
[url]PRIME and PRIME Synchronization - Linux - NVIDIA Developer Forums
to get an insight on how prime works.
I short, the intel gpu is driven either by xf86-video-intel DDX driver which is developed by intel or by the modesetting driver which is developed by Xorg and comes bundled with the xorg-server. Those paths nvidia-378 and nvidia-378-prime are non-standard and introduced by Ubuntu and their prime-select infrastructure. So you should really ask there as only someone who uses Ubuntu and knows how their prime-select works in detail will be able to help you. I, for an instance use Gentoo and don’t have anything like that so won’t be able to reproduce anything.