Not quite. The 1080i means there are 1,080 horizontal lines, but only half of them are shown at any given a time, and a moment later, the other half are active. Of course the swap happens so fast and so often, your eyes generally can't see it. The 'p' in 1080p stands for progressive, as opposed to interlaced, as demonstrated by a 1080i signal. A progressive signal shows all 1,080 lines at a time (or in the case of 720p or 480p, 720 and 480, respectively). This is supposed to help clean the picture up where there is heavy motion on screen.
Personally, I can't tell much of a difference.
Regarding 1080i meaning a resolution of 1366x768, that is simply inaccurate. It is likely an HDTV you are looking at is either a plasma or LCD with a native resolution of 1366x768, but still capable of accepting a 1080i signal. That means that somewhere between the signal source and the image you see, that 1920x1080 resolution image is being scaled down the screens native resolution (the maximum possible resolution capable on that set). You lose a lot of detail this way, but if you bought one, don't feel bad. Its what I have. My 37" Vizio has a resolution of 1366x768, so everything gets scaled to that resolution. At 37", an HD signal looks fine.
I would however, as long as the set is able to produce a heavily moving image with minimal artifacts, recommend an HDTV that has a native resolution of at least 1920x1080 and will display 1080p. It's the future, man.