Yeah so I definitely don't believe in (1). (1) would imply that closed-book exams should never be applied, ever, which I think is silly.

(2) is a pretty good summary of my position here.

If I want to test a student’s ability to, say, complete a complicated proof

So, I think this comes down to the question of what are exams good at, and what are assignments good at. If it takes longer than about 2 hours, it's probably just not a good topic for exam-like assessment. Exams, whether completely closed-book, completely open-book, or somewhere in between ("one page of notes" seems fairly common), specifically test someone's ability to work under time constraints, which in turn necessarily means it's also testing their ability to focus in addition to testing their actual understanding of the subject. Up to about 2 hours, that seems reasonable, but when you get too long, it starts getting unfair because the "focus" aspect starts outweighing the "understanding of the subject".

And if time isn't a constraint, and you allow them to work on it at their own pace over a week or more, well...that's just the definition of an assignment. In the modern world, I'll concede that assignments are very tricky. When I was in uni I regularly used Stack Overflow for some of my programming assignments, finding pre-existing answers to specific aspects of problems I had, in precisely the same way that today as a professional software engineer I often end up on SO. A couple of times in uni, I even asked questions on SO. Though these were not just asking the whole assignment on SO, but instead a narrow, focused problem I was facing. In my opinion, this should be considered acceptable.

What should not be acceptable is if someone puts the entire assignment up on SO and asks someone to solve it for them. I actually saw that once, when it came up as I was searching for help myself. They didn't get useful answers, thankfully.

And then there's a fuzzy line as to exactly how much help it should be acceptable to get, and I don't know how to draw that line.

Closed-book exams are useful because they test a student's ability to work under pressure and they test how well the student understands the information. Assignments are good because they test a student's ability to apply their understanding at a much deeper level when working on a larger problem.

But what's the value in a take-home exam, if we assume that the intent is to be closed-book but with effectively unlimited time? Presumably that means it's a problem roughly on the scale of an assignment, but they're not meant to be able to look up their notes, review the lecture material, etc.? I just don't understand what the point of that is. So even taking the practicalities of enforcing it out of the equation, I just don't think it's a worthwhile thing to do for a problem of such a scale. But when you do add in the practicalities, it becomes far clearer: much better to just let them use what resources are available and make it an assignment rather than an exam.

What it looks like when students copy code

For what it's worth, I've seen first hand that code copy-detection tools are honestly not actually all that great. Yeah, if they're stupid enough to just rename some variables and move some lines around, they'll get caught. But if you do even a moderate amount of refactoring—breaking some pieces into different functions, un-breaking-out some other material from methods into one big method, finding a set of variables that previously got used together and turning them into a class—even if the actual underlying steps the code is taking end up identical, the tools get fooled and the plagiarism is not detected. It's a classic case of how criminals (in this case, plagiarists—obviously not technically criminal) tend to be really stupid and that's the only reason they get caught.

Also, I learned the word “invigilator” today, so thanks for that

I'm actually not 100% sure on what "proctor" means, but based on how I've seen it used in this thread, I gather the two are the same? Proctor being American-English while invigilator is British- and Australian-English.