To make things simple, the Kornet label is, according to available sources, a whole "family" that includes the following sets:
9K128 Kornet-S - self-propelled tank interceptor with 9P161 launcher (BMP-3 chassis), 2 missiles ready to fire
9K128-1 Kornet-T[/url]9K128-1 Kornet-T - self-propelled tank interceptor with 9P162 launcher (BMP-3 chassis), 2 missiles ready to fire
9K129 Kornet-P - portable PTRK with 9P163 launcher
9K129-1 Kornet-P - portable PTRK with 9P163-1 launcher
9K131 Kliver - integration of the launcher into weapon stations and turrets of existing and prospective infantry fighting vehicles (BMP-1, BMP-2, etc.) not only of Russian production, which will allow to replace in these vehicles previous generations of PT missiles.
The 9K135 Vesti - a self-propelled light tank interceptor using a car chassis UAZ-3151, up to 4 missiles ready to fire (I assume this is an integration of the Kvartet module)
Kornet-E - export modification of the 9K129 kit (9P163M-1 launcher)
Kornet-D - modernized version with extended range (up to 10 km?), greater penetration and the ability to shoot even air targets with speeds up to 250 m/s using the 9P163M-2 launcher
Kornet-EM - export version of Kornet-D
Kornet-D1 - self-propelled tank interceptor with 9P162M launcher (BMD-4 chassis - for the prototype), or 9P163-3 launcher installed on the chassis of VPK-233116 / VPK-233136 Tigr with 2 x 4 missiles
The Quartet - Kornet/Kornet-E combat module with 9P163-2 / 9P163-2M launcher with 4 missiles designed to be mounted on light vehicles
Source: https://vpk.name/library/f/kornet.html
9K128 Kornet-S - self-propelled tank interceptor with 9P161 launcher (BMP-3 chassis), 2 missiles ready to fire
9K128-1 Kornet-T[/url]9K128-1 Kornet-T - self-propelled tank interceptor with 9P162 launcher (BMP-3 chassis), 2 missiles ready to fire
9K129 Kornet-P - portable PTRK with 9P163 launcher
9K129-1 Kornet-P - portable PTRK with 9P163-1 launcher
9K131 Kliver - integration of the launcher into weapon stations and turrets of existing and prospective infantry fighting vehicles (BMP-1, BMP-2, etc.) not only of Russian production, which will allow to replace in these vehicles previous generations of PT missiles.
The 9K135 Vesti - a self-propelled light tank interceptor using a car chassis UAZ-3151, up to 4 missiles ready to fire (I assume this is an integration of the Kvartet module)
Kornet-E - export modification of the 9K129 kit (9P163M-1 launcher)
Kornet-D - modernized version with extended range (up to 10 km?), greater penetration and the ability to shoot even air targets with speeds up to 250 m/s using the 9P163M-2 launcher
Kornet-EM - export version of Kornet-D
Kornet-D1 - self-propelled tank interceptor with 9P162M launcher (BMD-4 chassis - for the prototype), or 9P163-3 launcher installed on the chassis of VPK-233116 / VPK-233136 Tigr with 2 x 4 missiles
The Quartet - Kornet/Kornet-E combat module with 9P163-2 / 9P163-2M launcher with 4 missiles designed to be mounted on light vehicles