The train does have free WiFi, but my experience with it is pretty bad. You're also often remote enough that you can't use a cellular hotspot, so I wouldn't ever travel on Amtrak assuming you'll have a reliable Internet connection the whole time.
It doesn't.[0] Nothing about Starlink inherently requires the dish to be stationary. Starlink is being integrated into several airlines[1][2] over the next year, and I'm certain other airlines are exploring Starlink as an option... the existing in-flight internet options are terrible, so Starlink could make a real difference there, especially over the oceans.
Proper support for moving vehicles probably benefits significantly from a dish designed for that, but airlines, trains, cruise ships, yachts... there are many markets where such a dish would be valuable, so Starlink would be pretty dumb not to be designing one that works well for that.
Hawaiian Airlines certainly isn't planning to just bolt the existing dish to the outside of the plane and hope for the best.
Current subscribers are only authorized to use their dish in one specific area. However, this has more to do with capacity planning than anything else. Starlink recently announced a slightly more expensive plan that allows the user to roam from area to area without contacting customer support to reauthorize them for each new location. And as someone else pointed out already, the are definitely working on providing service to customers on boats and planes and other moving vehicles. I doubt they will ever make a dish designed to be mounted on the roof of a car, but larger multi–passenger vehicles are an obvious move.
Incidentally, Starlink dishes showed up on the landing barges used by SpaceX some time ago; they use them to stream live video of the landings now.
Mostly coding offline, but I did have a cellular hotspot that managed 4G LTE for most of the trip. I didn't investigate the train's WiFi since my needs were covered.
There was one state I didn't seem to have any cell service at all, Wyoming maybe? It seemed very similar to driving cross-country via i80 in terms of cellular coverage, with a few additional dead-zones where only the train goes. But in those spots the scenery is so beautiful you won't miss the cell network.
It’s not important whether the train has a WiFi router or not. What is important is what the WiFi router connects to. I don’t know what is common on European trains, but in the US they connect to whatever cell–phone network is closest. That is ok in urban areas, but in the US there are wide stretches of countryside where nobody lives. Somehow the phone companies never end up building cell towers in the middle of the deserts or on top of the Rockies. The train could easily be 50 or a 100 miles away from the nearest cell–phone network at any given time.