I usually put it in the form "the speaker is choosing to present the past event as relevant to the present".
But either way, the particular relevance (or connection) can vary. Some examples are:
a state which continues to the present ("has lived");
an action which has a continuing effect in the present ("has written");
an action which is so recent that it seems still relevant (No clear example in the ones you give, though "has finished" might be in this category);
(especially in the negative), regarded as taking place in a period which extends to the present ("haven't seen him").
In many cases, the speaker can choose the past or the present perfect, depending on whether they want to present this "present relevance": So "he finished his homework" is not relating it to the present - this might indicate that you weren't talking about the recent past, but some earlier day; but it doesn't necessarily do so; whereas "he has finished his homework" almost certainly means that it was recent, probably today, so the homework being in a finished state is still relevant.
"Have been to" is a special case, and I don't think they should have listed it with the others. It is an idiom that means "went to at some time, and came back". The "present relevance" there is that the visit is seen as taking place in a time that stretches up to the present; but it could have been long ago. If you don't use the perfect, you don't get the idiom: "He went to Canada" does not imply that he came back: he might have, but you can't tell. "He has been to Canada" means that he went and he came back.
Edited in response to comments.