As a UK citizen, I will be flying to Vancouver (for which I will need an eTA) and then taking an AMTRAK train to Seattle. I will stay in Seattle for several days before returning to Vancouver and flying back to the UK. Do I need an ESTA for entering the US during this trip?
Not having an ESTA will involve you in a lengthy secondary process to get a paper I-94.
If you are [a citizen of a VWP country and] coming by land, you do not need to have ESTA authorization, however, you should return your green I-94W card upon departure.
on closer investigation it turns out that US CBP does not consider an international train to be a land crossing (presumably there is some confusion about how trains actually work).
Travelers crossing from Canada into the U.S. on Amtrak Cascades trains AND are NOT U.S. or Canadian citizens nor have a valid visa, MUST have an approved ESTA, otherwise a paper form CBP I-94 will be required. If you are a VWP (Visa Waiver Program) traveler, this applies to you. (Amtrak Cascades trains do not fall under the U.S. CBP “Land Border Crossing” definition.)
Tom W's excellent answer gives more details.
The accepted answer correctly quotes the U.S. Customs and Border Patrol advice, however nowhere in this advice does it point out that CBP does not consider trains to be land. What they actually mean is arriving by road.
I know this from first-hand experience having departed for the U.S. by Amtrak starting in Vancouver. There I was firmly chastised for not having completed an ESTA. Only when one of my friends questioned this was it understood that we had followed the instructions as we understood them, but our understanding was wrong.
The Amtrak website claims:
Amtrak Cascades trains do not fall under the U.S. CBP “Land Border Crossing” definition.
Travelers crossing from Canada into the U.S. on Amtrak Cascades trains AND are NOT U.S. or Canadian citizens nor have a valid visa, MUST have an approved ESTA, otherwise a paper form CBP I-94 will be required.
Getting a paper form I-94 is possible but it means delays at best, according to the CBP:
If a traveler would like a paper Form I-94, one can be requested during the inspection process. All requests will be accommodated in a secondary setting.
You don't need it, but when using the Train from Vancouver (though not the one from Montreal), it will exempt you from having to obtain a green I-94W form.
The Amtrak website makes too much of a deal from the requirement to have a paper I-94 form...
First of all technically there is no longer such a thing as a "paper" I-94 - all entry records have been fully electronic since 2015. The paper that's stapled into your passport when you cross the border has no more legal weight than a print out from the DHS website stating your current I-94 status.
Second, all visa holders (non-VWP eligible) need a "paper" I-94 as well if they don't have a valid one when they cross the border. Therefore all class A border crossings (which includes the train station) are well equipped for issuing one, so it's not a big deal. Just because the CBP officer is lazy doesn't mean he's right for chastising you.
Finally, I'm not sure why Amtrak claims rail crossings are not "land" crossings. If it was truly so VWP-eligible passport holders would not be allowed to board the train without an ESTA rather than offered the option of getting an I-94W on the spot. So I think there's some amount of confusion over how the VWP works on their side.