Overnight is the standard overnight wilderness permit. Overnight exiting Mt Whitney means you are going in on a trail other than the Main Mt Whitney trail, and you plan to exit on the Main Mt Whitney Trail. There is a separate and second permit required if you want to exit to Whitney Portal using the trail down from Trail Crest. The thinking is that the trail is so impacted, and that many people were hiking to Mt Whitney from the north and south, that they need to limit the number of people hiking down the trail. So if you select the "...exiting..." part, it is way harder to get the permit--they won't let you get the entry permit if the exit permit quota is already full.
Got it, "Exiting Mt Whitney" is basically a 2nd permit allowing access to that section of the Whitney Main trail, eastward only (even though it looks like you are buying a single permit on recreation.gov). Once the "Exiting Mt Whitney" quota is reached, it is unavailable for all Inyo trailheads. This quota looks to be 25 per day, is that correct? But it is actually based on the day you are starting, not exiting, right?
"Next 5 Days" button: It works for me, but I'm using a browser on a full-screen. I tried it on my iPhone, and it shows "next 4 days", and that works ok, too. Maybe try a different browser.
Actually I noticed later it seems to do the 5 days ok, until you get past the 6 month reservation window, then I get the 14 day behavior. Weird.