Just tested this now. Parked a character in Luna on Pacific. She had Show Names: All selected. After 14 minutes of just standing there, she got the warning about being logged off for inactivity and then was logged off a minute later. During that 15 minutes, a few things other character said showed up in her chat log. However, no names showed up there at all. This was also true after logging in and changing the option to Show Names: Approaching.
I double checked the Chat filters and there does not seem to be any way to get names of ANYTHING you encounter to show up in your chat log. Aside from the issue of whether or not this is what keeps you logged on while inactive, I think this is a deficiency when you're in a situation where someone (one or more) runs by so quickly you don't catch their name. There's no way to look up that information in your chat log.
I don't know if the chat logs can be saved yet. If they can, I'm sure people who roleplay and people who do any kind of event reporting will greatly miss having names saved in the chat logs.