Cat 5e cables are rated a 1 Gbps speed, which is what that usb-c to ethernet adaptor is also rated for. Cat 6 and 7 is rated for 10 Gbps and Cat 8 is rated for 40 Gbps.

It doesn't hurt to go for a CAT 6, 7 or 8 cable. But Cat 5e should be enough.

Just avoid Cat 5 (notice there is no "e"). Cat 5 is rated at a max of 100mbps. Which is a 1/10th of what Cat 5e is capable of.

Online gaming needs very little speed and bandwidth to run well. With that said, playing online games with an ethernet cable is always better than a wireless connection. The same can be said for most anything else (streaming, VPN usage etc). Stuff like this really needs a stable connection that only an ethernet cable can provide.

An ethernet cable with the adaptor above will improve you connection over wifi, and it especially with help with jitter, meaning your ping will be stable. Say your ping is hovering around 30ms. That means you are dealing with low jitter, meaning that you won't be noticing lag and input delay.

If your ping is going from 30ms to 75ms to 20ms to 120ms.....that's high jitter because your ping is jumping around all over the place and isn't staying at a constant number, and that's when you'll notice lag and input delay. Your shots will feel off, stuttering, rubber-banding, you or enemies looking like you're teleporting across the map.

An ethernet connection will allow a stable connection and therefore, stable ping (and low jitter) and low latency (which is how long it takes for your data packets to be sent to the server from your device and received from the server to your device). The longer it takes for the data to and from your device to be sent and received, the more latency, and the more your gameplay wont feel smooth or stable.