No TV should be advertised as a 1080i display unless it actually is a 1080i display.
Many CRT rear projection televisions and CRT standard tube type televisions are 1080i native displays. They accept 1080i and display it as a true 1080i signal. Many of these displays do NOT accept 720p HDTV which is a very poor thing for any HDTV not to be able to do.
But, when you are talking about a digital display - including DLP, LCD, and LCoS, then you are talking about a progessive display. The ONLY number that matters is the native resolution of that display.
1920x1080, 1365x768, 1280x720, 1024x768, and 854x480 are the most common resolutions you will run into with digital displays for home theater.
It is important to know that 1080i DOES NOT EQUAL 1080p. But, at 60hz there is almost nothing that is 1080p out there. Since movies are 24hz, they should be handled at 1080i or 1080p and look identical. But, this is rarely going to be the case due to poor deinterlacers in many displays.
The bottom line answer to your question is that you have things correct in your head already. The ONLY displays that are native 1080i will be CRTs. Digital displays are all progressive and a 1920x1080 digital display may only handle a 1080i signal and not a 1080p signal, but the display itself will still have to convert it to 1080p to show correctly on screen.