Unfortunately the general confusion highlighted above is extremely common where ticket machines are involved, and I am sure it is a big money making scam by the TOCs and their beloved ticket machines.
The number of people I see travelling with "Anytime" tickets they have purchased at a ticket machine, when an "Off Peak" ticket would have been perfectly valid and significantly cheaper, is amazing. I started asking people and trying to advise them once, but there are simply so many that if I did that then I wouldn't get anything else done.
It seems to revolve around the OPs original conundrum - someone sees the words "Off Peak" on the ticket machine and then assume they can't use it as they are travelling in what they think is a peak time, and therefore the ticket isn't valid. Now as we all know the Railways idea of simplicity is far from it, and therefore there is no easy definition of "peak" or "off-peak". The average punter, forced into using the aforementioned ticket machines gets confused by the "Anytime" / "Off Peak" boundaries and plumps to buy the (normally) vastly overpriced Anytime ticket to avoid problems.
Stangely however, on the geniune Peak time trains the reverse seems to be the problem. People appearing with "Off Peak" tickets they have just purchased from the machine when they are not valid. The normal excuses of "The machine never told me" / "I've just brought it now so it must be valid" / "That man on the platform said it was ok" are wheeled out, but as in most of these early morning cases I think that lots of them are commuters or regulars trying it on I will excess them up to the correct peak fare, explain the restrictions, and ask them to contact Customer Relations should they have any concerns. I have checked before when I had time to kill, and the ticket machine did display the relevant restrictions when the buttons were pressed, so that excuse normally falls on deaf ears.
These situations are however going to get more common. As more and more offices close, and more and more machines appear, the cases of people being overcharged will no doubt increase.
Most Recent example : I passed through Leeds City (which is extremely busy) last Wednesday evening to meet a friend for food and beer. As I passed through the ticket office I noted that at 1700 on a Wednesday evening there was only 1 - yes 1 - ticket window open with an absolutely enormous queue. The huge bank of ticket machines were busy , but by no means were all in use. When I went back later on the queue had gone down a bit and 2 windows were now open - but by this time the travel centre had closed so all enquiries were now coming here as well! This is no doubt a deliberate ploy to get people to use machines and thus spend more money!
Example 2 : My Uncle visits occasionally from Salisbury to Yorkshire. He always used to go and visit Salisbury travel centre and arrange tickets - and he would always appear with some ludicrously cheap advance purchase tickets. SWT have now closed the travel centre and reduced the number of ordinary ticket windows that are open. But never fear, they have put in lots of nice machines. My uncle went to try and purchase advance tickets twice recently. On 1 occassion the queue was enormous and after a long wait he gave up and went home. The second time he went a similar situation, but apparently there was a member of staff "Queue Busting" and ushering people over to the machines to buy tickets. He asked about buying advance tickets and was alledgedly told that they don't have time for that any more unless it is very quiet and he should go home and buy them on the internet. It seems he did go home, but as he is no techno wizzard I dread to think what tickets he has ended up with. I did try and get him to use the East Coast site and gave him all the details and what to do, but it sounds like he has paid more than I think he should.
Multiply all these situations across the country, and the TOCs purses are onto a winner - reduce ticket offices and hence staff costs. Force punters to use unfriendly machines. Punter gets confused by said machine and variety of tickets and ends up paying more than they should. Costs down and ticket revenue up - KerChing!!