Thanks for raising these important questions. In fact, I was planning to write a blog article about a related issue anyway.
So what is the meaning of a sentence supporting an
argument, when it is not a premise?
Let's first make this question more precise: You can use two sorts of relations (arrows) in Argunet: The sketching arrows do not presuppose any kind of "model of argumentation", so let's not talk about them. I think what you mean are the "real" relations.
This is important because the case you are describing is only possible with sketching arrows. If you are defining logical relations between sentences and arguments this case is impossible, because by definition a sentence supports an argument, if this sentence or any equivalent sentence is used as a premise in this argument (the same is true for your second case).
This seems redundant for me.
Is this redundant? (It is obviously not redundant if you are using several equivalent sentences ("The Beatles are better than the Rolling Stones", "The Fab Four are better than the Rolling Stones"), but let's put that case aside) Maybe it can be in some cases, but that's up to your reconstruction (we are trying to make Argunet as theory-independent as possible).
Let's just look at some cases in which sentences are very useful indeed:
Argument-to-argument relations are based on conclusion-premise relations. But there are important premise-to-premise relations and conclusion-to-conclusion relations that you might want to visualise as well. By adding a premise or a conclusion as sentence to the argument map you can visualise these relations.
So for example if two (maybe otherwise very different) arguments share a premise like "Human rights have to be protected at all costs" you can visualise that by inserting the premise as sentence to the map. Two green arrows are then automatically added from the sentence to the two arguments. Or it might be important to visualise that there are several arguments that justify the same sentence (maybe a central thesis of the debate).
This is the reason we use sentences in our model of argumentation. We could do without sentences as independent elements if we would a) use more kinds of relations (yellow, blue, ... arrows) or if we b) would broaden the definition of the current ones (a green arrow from argument x to argument y can either mean that .... or that ... or that ...). Both options aren't viable alternatives. We are quite proud of our minimalistic set of elements (against a) that are nevertheless defined in a very precise way (against b).
Sentences are used by arguments but they are at the same time independent of them. They can occur unaccompanied and they can have many different associations. To visualise these without using sentences as visual elements would often lead to a chaos of cross-cutting arrows.
A last point I will elaborate on in my blog-post: Sentences allow you to create argument maps that are very similar to those many people know already know from software like Rationale. If you want to create such a map, you just have to "unpack" all arguments completely by adding all premises and the conclusions as sentences to the map. In version 1.0.1 we have added two context menu options for this purpose.
But we do not really think that it is always a good idea to "unpack" all arguments in this Rationale kind of way (especially not if your argument map is fairly complex). It is one of the advantages of Argunet argument maps that you can decide on a case by case basis which sentences are important enough to be visualised in your argument map. This is not so much a question of your model of argumentation as a question of your style of reconstruction.
Wow, this has gotten longer than I planned it to be... sorry. But I hope this answers your questions about the theoretical status and practical use of sentences in Argunet