Let me try to explain what material implication is. There are three basic kinds of implication that we use. There is a causative implication, when we mean that the antecedent causes the consequent. There is logical implication, which denotes class relation, such as the Socrates syllogism, and there is definitional implication, such as “if a polygon has only three sides, it’s a triangle.” Material implication is none of these by themselves - it extracts the common partial meaning of each. Like each of these three types of implications, however, it is a truth-functional relation. It just has no use when we cannot assign truth functions. But it is extremely useful when we can.
But notice that those three basic types of implications, which are accounted for by MI, do require a connection of some type between antecedent and consequent - by definition. Therefore, some connection is part of their common partial meaning, and is therefore also included in MI.
Now, most of the time, when we are using implications, we are probably using causal implications, which are never used in mathematics. For this reason MI is perfectly useful in mathematics, and no mention of a logical or definitional connection need be made, because this relation is already built into mathematical terms. I don’t know about your third citation, but it seems clear that in the first two, given that they are mathematical treatments, the definitions they use are probably adequate. You could clear this up with quotes.
But let’s consider one type of conditional that poses a problem similar to the one you state. Let’s say that the Green Bay Packers have already clinched their division (which they probably have, I haven’t checked). And I say, “If the Packers win their next game, then they will win their division”. And they win their next game. Both simple statements are true, and the implication is formally true within the definition of MI, but there is no logical, definitional nor causative connection between them. Is this really an implication? Both parts can be verified, can have a truth value assigned (at least after the next game). In ordinary language, this takes the form of an implication, but it is not an implication. No one would take it as such, if they knew the facts involved. Any more than they would consider “If I eat chicken tonight, then the Packers clinch the division”. Yet it still follows the rules of MI.
This is so because MI is an abstraction created to account for implications with different meanings. MI does not capture the actual and full meaning of any given statement. It can, then, result in what may strike us as silly implications. But that’s the price of the simplicity and clarity that MI provides when we do make actual implications that have meaning in the first place. And the reason is this - MI is a rule to used to show the validity of an argument, and not to ascertain the truth of its component parts. MI is not a truth-generator, but a validity-generator. It shows only valid argument form, and not truth.
So why do we assign truth values to the component parts of a conditional? Well, we don’t, always. When we cannot, modal logic seeks to fill the gap. MI is used when we can assign truth values. But modal logic can be used in these circumstances as well. In the case of the Packers, the conditional may actually be “modal”, if we do not know all the facts about the playoff picture in the NFL this season. Modal logic just has a different use, and that difference may depend on the person using the facts, and not just the facts themselves. This is an oversimplification , in that modal logic would need some slightly different expressions, but my point is that MI’s sometimes silly consequences is not due to MI itself, but to its use at times.
