I don't think there's any real distinction between ability and possibility in these cases. Either way you interpret can, the sentences are true.
English modal verbs are sloppy. Each serves many different purposes and meanings, sometimes simultaneously. They lack the precision to make such a fine distinction. When you need to make a distinction like that clearly, you must add some extra words to the sentence to spell it out precisely.
You can* see the same ambiguity in will. The older, primary meaning of will expresses mental determination or choice in the present tense: "Do as you will." Will also, much more commonly today, just indicates the future tense: "What will happen tomorrow?" So, which meaning is intended in the following sentence?
I will win!
You could understand it as expressing the speaker's present mental determination to win, or you could understand it as stating a prediction that the speaker will win. Really, though, the verb doesn't distinguish. The speaker sort of means both, and a listener sort of hears both.
* Now that does that can mean possibility or ability? Hmm, mostly ability, but, uh…