Perhaps this content from Wikipedia will help sort out any remaining confusion.

One use of the term “display resolution” applies to fixed-pixel-array displays such as plasma display panels (PDPs), liquid crystal displays (LCDs), digital light processing (DLP) projectors, or similar technologies, and is simply the physical number of columns and rows of pixels creating the display (e.g., 1920 × 1080). A consequence of having a fixed-grid display is that, for multi-format video inputs, all displays need a "scaling engine" (a digital video processor that includes a memory array) to match the incoming picture format to the display.

Note that the use of the word resolution here is a misnomer, though common. The term “display resolution” is usually used to mean pixel dimensions, the number of pixels in each dimension (e.g., 1920 × 1080), which does not tell anything about the resolution of the display on which the image is actually formed: resolution properly refers to the pixel density, the number of pixels per unit distance or area, not total number of pixels. In digital measurement, the display resolution would be given in pixels per inch. In analog measurement, if the screen is 10 inches high, then the horizontal resolution is measured across a square 10 inches wide. This is typically stated as "lines horizontal resolution, per picture height;"[1] for example, analog NTSC TVs can typically display about 340 lines of "per picture height" horizontal resolution from over-the-air sources, which is equivalent to about 440 total lines of actual picture information from left edge to right edge.[1]

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That's a bit of a stretch. In this case even the upstream article introduced the concept as "TV with 16 times the resolution of HDTV." You are technically correct of course, but you have to acknowledge that the common use of the term and the technical use of the term "resolution" with regards to televisions have evidently diverged. 8-bit vs. 16-bit color having half the number of colors is just patently absurd.

You're right, I meant 720/24p. The use of the word "broadcast" was also careless on my part in this situation.

As you've pointed out it really comes down to terminology.

Pixels per inch depends on the "resolution" of the display standard, and the size of the display. a 40" 1080p display and a 90" 1080p display have the same number of pixels, but wildly different pixel per inch values.

None of these standards say anything in terms of pixels per inch - just number of pixels overall. "resolution" in this context can only mean the total number of pixels. 4k has 4x the number of pixels that 1080p has, and 8k has 16x the pixels.

It really depends on what it is that you are encoding. p will do a lot better if you have a lot of motion in the video.

It depends on how you want to describe resolution. The standard does not dictate anything about how dense the pixels must be just how many of them there are. In this case, there are 16x as many pixels as there are in a 1080 signal.

They already do to some extend. Sports for example usually require more bandwidth to deliver acceptably due to the higher amount of motion in the video.

It's highly unlikely that you are watching much of any 1080i/p content over your cable/satellite subscription in the US. Most of it is distributed as 720

is 8k and the equivalent of 4x 4ks, which are themselves the equivalent of 4x 1080s. It's confusing because both 4k and 8k are covered by the UHDT standard.

Incorrect, 7680x4320 IS 8k, 4k is 3840x2160

The issue is the number of pixels that you have to put into a given area of screen - that is what is more difficult.

Ok so semantics perhaps. 16x the number of pixels - depends on how you want to describe "resolution." One common definition is the amount of detail that the image holds, in which case, 8k is 16x the resolution of 1080.

You are missing something, it is 16x the resolution. 4k is 8x the resolution.