And as far as I understand, isn't NVMe SSD just a "different name" for PCIe SSD? PCIe being the protocol (that's already being used for graphics cards), and NVMe the standard for SSDs to understand that protocol.
There's more to it than that. NVMe is a higher layer technology than PCIe.
SATA drives connect to the host system over a SATA PHY link to a SATA HBA that itself is connected to the host via PCIe. The OS uses AHCI to talk to the HBA to pass ATA commands to the drive(s).
PCIe SSDs that don't use NVMe exist, and work by unifying the drive and the HBA. This removes the speed limitation of the SATA PHY, but doesn't change anything else. The OS can't even directly know that there's no SATA link behind the HBA; it can only observe the higher speeds and 1:1 mapping of HBAs to drives. Some PCIe SSDs have been implemented using a RAID HBA, so the speed limitation has been circumvented by having multiple SATA links internally, presented to the OS as a single drive.
NVMe standardizes a new protocol that operates over PCIe, where the HBA is permanently part of the drive, and there's a new command set to replace ATA. New drivers are needed, and NVMe removes many bottlenecks and limitations of the AHCI+ATA protocol stack.
Not quite. PCIe is just the bus transport protocol. The PCIe SSDs still need a protocol to describe data operations to and from the OS to the Drive. Currently, they use the same protocols as those designed for HDDs, and those protocols make a lot of tradeoffs and assumptions about disk access time. The reason NVMe is exciting is that it is a brand new data protocol, designed with low latency SSD style storage in mind. We're already seeing speeds much higher than existing PCIe SSDs can manage (3+ GB/s!)
Most current PCIe SSDs behave to the PCI bus & operating system like an AHCI controller with a SATA SSD attached to it. Advantage of this is that it's well supported by firmware (BIOS/EFI) and operating systems (stock drivers). Such SSDs inherit the limitations of that architecture, which was designed for spinning platters. NVMe is a fresh start in that regard.
