A little more detail. After several stabs at this, the USB creators have landed (so far) on this naming system. ALL USB3 system now are called USB 3.2 Genxxx, and all previous systems are compatible with a few caution notes. Original USB 3 and USB 3.1 are now all part of USB 3.2 Gen1. USB3 cables with the old-looking Type A connectors and sockets all still work for all of this. But NOTE (you may know already) that USB 3 cables are DIFFERENT from USB 2 because of the number of wires and contacts needed for the faster data rates. The differences between the Gen numbers in USB 3.2 is in the hardware at the end of the cables.
USB 3.2 Gen1 has a data transfer max rate spec of 5 Gb/s
USB 3.2 Gen2 has a data transfer max rate spec of 10 Gb/s
USB 3.2 Gen2x2 has a data transfer max rate spec of 20 Gb/s
A cautionary note on data rates and cables, though. There are cheap cables and high-quality cables, and some overpriced cables and some really good cables at reasonable prices. It's hard to tell by sight, so look for good evaluations. There is a genuine question whether the classic (but upgraded to USB 3) Type A connector is capable of data speeds over 5 Gb/s reliably. So it is recommended (strongly so for Gen 2x2) that you use cables with the new Type C (much smaller) connector system at the higher speeds. In YOUR case, OP, you can connect your Type A case sockets to either Gen1 or Gen2 mobo headers. It MAY be, once such devices are available, that you don't achieve the data rates over 5 Gb/s from the Gen 2 headers if the Type A sockets are not quite up to it.
May be? Why did I say that? These USB 3.2 systems have been designed and deployed now intending that their max data rates are already much higher than any device currently in use. As faster future devices are deployed, they will not be limited by the USB 3.2 communication system capabilities for some time to come.
Simple devices like keyboards and mice are slow, and can be handled easily by the older USB 2 system (max 0.48 Gb/s)
Faster data devices like SATA III HDD's certainly need the speed of Gen1. For mechanical hard drives (spinning disks and moving heads), the mechanical parts cannot move fast enough to deliver data any faster than about 2.5 Gb/s, which still does not get to the SATA II limit. Many modern SSD's can exceed that SATA II limit (3 Gb/s), but still are well within the capabilities of the new USB 3.2 Gen 1 limit. The very fastest SSD units may exceed that and benefit somewhat from the Gen 2 increased ability, but they are not close to the 10 Gb/s limit. There is nothing on the current market over that data rate.