To know which HDMI 2.0 / 2.1 or DP 1.2 / 1.4 cable you need, you'd need to know whether your screen is 4K or other? But it's highly likely that your problem is due to the fact that you need to activate overclocking in the screen options (buttons under the screen, not in Windows) to go from 144hz to 165hz. I have this problem in 4K, I have to activate overclocking to go from 120hz to 144hz in DP 1.4 I think.

Every HDMI 2.0 cable and larger is capable of 165hz. Including the one you bought.

Your DP 1.2 can also do this according to the specifications. Is your graphics card also capable of 165hz?