This is a critical distinction. As a concrete example, here's how voting worked in a scheme I once read about. On any one ballot, the order of candidates was randomized. Then the way the scheme worked was that after voting, the voter tore off the candidate positions (but not their vote) and threw it away in a huge pile of them, burned it, or whatever. (Made it so that someone couldn't come behind them and figure out their position list, essentially.)
Later, after the votes were tallied, the voter could verify that their ballot was (1) counted and (2) counted towards their chosen candidate. But crucially, all they could verify was that the vote counted towards position 1, or position 2, or position 3, ...
The point is that since the voter couldn't prove to a coercing party that the position they voted for was (or was not) the candidate the coercer wanted them to vote for, they were immune to coercion. They could prove that they voted for position 2, sure. But which candidate was at position 2?
The voter knows the truth because they saw the position list. However, until we have mind-reading technology, a coercing party could only take the voter's word.
I'm not following how the counting is done. If all the counter has is a ballot with position 2 checked and the corresponding candidate name torn off, how does that vote get tallied to the proper candidate?
Later, after the votes were tallied, the voter could verify that their ballot was (1) counted and (2) counted towards their chosen candidate. But crucially, all they could verify was that the vote counted towards position 1, or position 2, or position 3, ...
The point is that since the voter couldn't prove to a coercing party that the position they voted for was (or was not) the candidate the coercer wanted them to vote for, they were immune to coercion. They could prove that they voted for position 2, sure. But which candidate was at position 2?
The voter knows the truth because they saw the position list. However, until we have mind-reading technology, a coercing party could only take the voter's word.