But 2K and 4K do refer to the horizontal resolution. There’s more than one resolution that’s referred to as 2K, for example 2048 x 1080 DCI 2K, but also 1920 x 1080 full HD, since it’s also almost 2000 pixels wide. The total number of pixels is in the millions, not thousands.
For 4K some common resolutions are 4096 x 2160 DCI 4K and 3840 x 2160 UHD, which both have a horizontal resolution of about 4000 pixels.
It’s off by default, but activated when you end your search query with a question mark. That option can be turned off.