Your friend is correct; the AX200 (WiFi 6) and AX210 (WiFi 6e) are your only possibilities. The AX201 requires a system with CNVIO2 support, which wasn't supported until the Comet Lake/Ice Lake chipsets supporting 10th Gen processors (see https://en.wikipedia.org/wiki/CNVi for additional information).
Hope this helps,
P.S. The AX201 only supports WiFi 6. For WiFi 6e support, you need to use the AX210.
I am using AX200 with 8 Series motherboard and 4th gen i7-4770K processor (this is using PCIe x1 carrier card with USB connection for Bluetooth), so see no reason why the card will not work with 6th gen processors (Mobile or Desktop).
If your laptop provides only the older PCIe Half Mini Card connector, a third-party company makes a version of the AX200 card for this connector. I am usually reticent about these third-party adapters, but decided to give one a try. It worked like a champ (have it in similar 4th gen processor system using older carrier card). If you are interested, here is information about this card: https://www.newegg.com/derapid-mpe-ax3000h-mini-pci-express/p/0XM-02WM-00014.
Increases in Internet speed are going to be dependent upon the connections made possible (distance to router, interference from wiring, etc. in the direct path, etc.). In general, if you had a Wireless-AC 9260 or newer and upgraded to the Wireless-AX AX200, you aren't going to see much of an increase in (single-band) speed. If you had a Wireless-AC 8265 or older, on the other hand, then yes, you are going to see a significant increase.
Hope this helps.