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Yes.

While US Customs and Border Patrol says:

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. If you are arriving by land and departing by air, you should return your green I-94W to the airline representative prior to boarding your flight.

However on closer investigation it turns out that US CBP does not consider an international train to be a land crossing

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.)

(presumably there is some confusion about how trains actually work).

Tom W's excellent answer gives more details.

Not having an ESTA will involve you in a lengthy secondary process to get a paper I-94.

While US Customs and Border Patrol says:

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.

No.

US Customs and Border Patrol says:

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. If you are arriving by land and departing by land or sea, you should return your green I-94W to the airline representative prior to boarding your flight.

Yes.

While US Customs and Border Patrol says:

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. If you are arriving by land and departing by air, you should return your green I-94W to the airline representative prior to boarding your flight.

However on closer investigation it turns out that US CBP does not consider an international train to be a land crossing

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.)

(presumably there is some confusion about how trains actually work).

Tom W's excellent answer gives more details.

No.

US Customs and Border Patrol says:

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. If you are arriving by land and departing by land or sea, you should return your green I-94W to the airline representative prior to boarding your flight.

No.

US Customs and Border Patrol says:

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. If you are arriving by land and departing by land or sea, you should return your green I-94W to the airline representative prior to boarding your flight.