Yes, k stands for thousand. Historically, 4k meant 4096 pixels wide in the context of digitised film or digital visual effects, and 2k meant 2048 pixels wide. TVs ended up 1920 pixels wide, which is "close enough" to use the same term, 2k. I think "4k" is used for marketing TVs as it's easier to remember and say than "2160p", so now we mix the terminology.
Its a loophole. UHD4k is what tv advertiser used to dilute meaning and not get sued. I believe the definition has been diluted to the point 4k is expected to be UHD.