I don't know if you're confused or getting into a semantic argument or what, but let me try to clarify the original explanation.
Like I mentioned before, most stations broadcast in 720p60 or 1080i60.
In both cases you can push 24FPS, 30FPS and 60FPS content through.
Obviously there's judder thats introduced with 24FPS unless you have a 120hz panel which is smart enough to recognize the 24FPS content and display it correctly. On a 720p broadcast it's already a progressive signal. On a 1080i60 broadcast, it comes in as interlaced but all of the information to reconstruct it as a 1080p video is there. Modern TV's are able to apply some processing to get it to it's original progressive state-- there's no information lost. So even thou it's a 1080i60 broadcastyou're essentially seeing a 1080p24 brodcast on a 1080p panel.
For 30FPS it's pretty much the same deal except there's no judder introduced since 30 fits into 60hz evenly (every frame is displayed twice).
For 60FPS on a 720p signal, it's straight forward, every frame is a unique progressive image.
For 60FPS on a 1080i signal, the way it works is that there are 60 frames of information, but the even frames only contain image information for horizontal lines 1, 3, 5, 7, ... 1077, 1079 and the odd frames contain image information for horizontal lines 2, 4, 6, 8, .... 1078, 1080. On an old school CRT HDTV what would happen is that it literally would display only the odd lines one frame, and then the even lines the next frame. Since the lines that information was being displayed on were alternating so fast, it would trick your brain into thinking you were seeing a progressive image and it would look pretty good. On a fixed pixel display (like LCD) they can't really do that-- you'd see way too much flicker from the image and it would look bad. So instead they have algorithms which for the even frames will fill in the information for lines 2, 4, 6, 8, .... 1078, 1080 based on the previous frame's lines of 2, 4, 6, 8, .... 1078, 1080, the current frame's lines of 1, 3, 5, 7, ... 1077, and the next frame's lines of 2, 4, 6, 8, .... 1078, 1080. And for the odd frames it will fill in the info for 2, 4, 6, 8, .... 1078, 1080 using a similar technique. Yes its interpolation going on, but its not interpolating a 30FPS signal into a 60FPS progressive signal. Its interpolating a 60FPS interlaced signal into a 60FPS progressive signal. And what your eyes see on your TV is literally a 60FPS progressive signal.