A physical polling station prevents this by ensuring a) the voter is not documenting the vote and b) no one else is documenting the vote. Neither a) nor b) can be guaranteed with online voting. It is extremely hard to provide PROOF to someone you voted a certain way in a physical voting situation. It is easy to SAY you voted a certain way, but that doesn't have to be true.
A physical polling station does nothing to ensure that the voter is not documenting their own vote, nor was it designed for this purpose. It's trivial in the modern era to take out your phone and film yourself voting, from beginning to end, inside the booth. Whether you throw some tantrum and manage to get your vote changed, or edit the video footage later, is your own business of course. Your peers pressuring you into demanding "proof" is just as much a problem with paper voting as it is with any other form.
The more important point, though, is this: If you don't want anyone else to see you voting on your smartphone, you can go hide in a closet and vote. If you do, you can always register your vote "in public" and then change it later. If you think someone is going to hold you at gunpoint and stare directly at your phone for the entire duration of the voting period - which can be as long as a WHOLE MONTH, considering how vote-my-mail ballots already work in this country - then you have much, much bigger problems than your ability to vote being tampered with. You are the victim of a kidnapping and the police should be out looking for you.
If you're especially paranoid I suppose the voting software could implement a "no take-backs" feature where you can lock in your vote, so even if you're kidnapped near the end of the voting period, you can't be forced to change it. Then the kidnapper has to simultaneously abduct enough people to sway an entire election the SECOND the polls open, then have enough coercive power with them - threat of imminent death for example - so that they don't just refuse to vote altogether. Again, if you live in a city where this can happen, you have bigger problems.
Same deal with the hypothetical Texas church: If your church locks you in and compels you to vote a certain way on pain of excommunication or whatever, you have much bigger problems at hand. You should be videotaping that and going to the feds with it. Sadly, if you're a member of such a church, you probably think the feds are an agent of Satan anyway. Properly implemented encrypted online voting is not going to influence this, since this sort of ugly fraud is just as possible with absentee ballots and voting-by-mail already.
(Note that this scenario is pretty damn out-of-wack. In many towns, the church is trusted as non-political enough to double as an official polling place.)