Leaving aside the question of whether it's realistic for the system to depend on non-voters checking their vote's status - what would be the procedure if someone claimed they were wrongly shown as having voted?

After all, it's easy to imagine a situation where certain groups might want to manufacture evidence of voter fraud.

In Pennsylvania at least, that is handled by the filing of a provisional ballot followed by a court system. One challenge to that solution is that there are hard limits on how long it can take for the election to resolve, so it is possible for that window to close before an investigation on an allegation of voter fraud to complete, and if it cannot be proven that that provisional ballot should be counted, the ballot is discarded.

The recourse should a person's vote be stolen that way is proof that the vote was stolen followed by punishment for the thief, and that punishment can be fairly severe because vote tampering cannot be taken lightly given the impact it can have on the process itself.

What elections you vote in is, in many cases in the US, pretty much public information.

Is there an officially recognised category of spoiled ballot paper in the US like in the UK?

Yes. I’m a poll inspector.

Spoiled ballots go into a separate lockbox. At the end of the night everything is tabulated (winners, losers, provisional ballots, spoiled ballots) and posted at the voting precinct. The spoiled ballots box is sealed and taken to the election office to be inspected. This is for 1 state. Every state manages elections differently.

BTW, if you can sign up to be a poll worker. It’s a thankless job that takes 1 day a year. Most poll workers are retirees and I was one of the youngest there.