An example of "general past possibility" would be:
"In Roman Britain, you could get from what's now Canterbury to what's now St Albans by travelling Watling Street."
As opposed to specific past possibility:
"I could have bought it for half the price yesterday!"
It is common usage, even if incorrect according to some grammars, to use the same syntax as the general possibility for a specific one:
"Yesterday I could buy it for half the price!"
We're actually seeing different senses of could here. In the present (or future) tense, could indicates a sort of hypothetical, where other languages might use a subjunctive form, perhaps. It is used to make suggestions or raise possibilities.
"What shall we do today?"
"We could take the train up to the lakes and do some walking."
It doesn't communicate a strong desire, though tone may indicate some desire. It can also indicate a possibility that one might need to think about.
"Where is she? She said to meet in the library at 3pm!"
"She could be at the other entrance."
Now, when we want to put a hypothetical in the past, we apply that same sense of could to the present perfect:
"Why am I slogging through all this coursework? I could have taken a gap year and be sunning myself on a beach in Thailand."
Used that way, it is expressing a hypothetical about things having been different. It is known not to have happened, but it could have happened.
"Where is he? He said to meet in the library at 3pm!"
"He could have gotten sidetracked. You know what he's like."
That is an unknown hypothetical in the past. You don't know whether it happened or not.
But when we use could in sentence like "in Victorian England, people could die of diseases that we can easily treat today", it's not the same sense of could as above. Let's talk about can.
Can is a modal verb that indicates possibility, rather than a hypothetical. It means be able to, and sometimes you have to resort to be able to because can doesn't exist or work in all tenses. So,
"I can touch my toes!"
"I will be able to pay for next week."
But what if you can't touch your toes now, but you were able to in the past? Well, then it becomes:
"I could touch my toes last week..."
Here, could is a past form of can. It means, roughly, "was able to". It says nothing about whether you did or not, merely that had that ability.
The two senses of could are closely linked, no doubt. It probably tells someone more educated in philology than I a great deal about how our language developed from its forebears. But now these two sense are quite distinct - if potentially confusing.