"Trail runners" are basically re-inforced running shoes. Many people use regular running shoes like new balance for example(NB also makes trail runners). Trail runners will have the flex and cushioning for running/walking just as a regular running shoe, but will offer a little more support, and are better suited for more rugged conditions. They are also low cut like a regular shoe. Also, I imagine the tread would be different because of the different surfaces, but I don't really know about tread. I like the Montrail Hardrocks and would reccomend them if you are looking for a trail runner.
A "light weight boot" would basically be that...a light weight boot. It would have the ankle and foot support of a regular "boot" but not the heavy materials(such as full grain leather). They will be a bit heavier than trail runners and quite a bit heavier than a regular running shoe.
I had to look up the definition of a "cross training shoe" but I came up with this: Cross trainers are basically an all-purpose sports shoe. "Cross-training shoes meet the basic comfort, cushioning, stability, and durability requirements of many popular sports and activities...". If you like to bike, run, play basketball, play tennis, etc. then they might be a good choice, but for JUST hiking or JUST running they don't have enough cushioning or flexibility. They are heavier than running shoes and not recommended for runners.