1080i is interlaced; 720p, 1080p, and 2160p are progressive, i.e. non-interlaced. My understanding is that most commercial Blu-rays are progressive encodings, broadcast may be different. (1080i was common back in the day.)
Interlaced had the advantage of looking better for fast motion, such as in sports; for example, 480i with 60 fields/sec would look better than 480p with 30 fields/sec, at equivalent bandwidth.
The purpose of interlace is to double the frames per second and capture motion better. The downside is some vertical resolution loss and some odd, but rarely seen, visual artifacts.
Doesn't interlacing alternate even lines from one frame with the odd lines from the next, thus showing 50% of each image? Which would make it a ghetto crossfade as described.
Not really. On a CRT monitor, the odd lines fade before the even lines begin lighting up, and they are on different physical locations on the monitor. Recombination of the two images to improve resolution occurs entirely in the human vision processing system.
It's worth noting that this takes advantage of the human vision system's features. When an object moves quickly across your vision (for example, when turning your eyes quickly down a field), there is resolution loss that your brain makes up for, And when focusing on a still object, vision allows you to see more resolution.
So when a player runs across the camera view, your brain correctly processes the lower resolution and higher motion resolution. And when a player is still, your brain correctly processes the improved resolution.
Interlaced had the advantage of looking better for fast motion, such as in sports; for example, 480i with 60 fields/sec would look better than 480p with 30 fields/sec, at equivalent bandwidth.