Dear Paul,
First, though, I would like to underline something that Steve Lack (almost) said, namely that the category in which you index your components, and therefore also the one in which you define connectedness, need to be EXTENSIVE, ie their coproducts should be disjoint, and stable under pullback, and the initial object strict.
Maybe we've over-done philology recently, but "component" means "putting together", where we expect the parts to cover the whole (coproduct), without overlapping (disjoint), to be distinguishable (like disjoint union, but unlike addition and disjunction). The modern notion of extensivity, in which Steve had a part, captures this idea very neatly. The equivalence between definitions of connectedness based on 1+1 and on X+Y surely depends on stability under pullback, and the requirement that the choice between left and right be unique surely requires disjointness. Maybe a close study of Marta Bunge's work on abstract connectedness would clarify this.
In my now obsolete 1966 thesis, the context was that of a category with finite limits and finite coproducts, but I had not assumed therein that coproducts should be disjoint and universal. With these assumptions, the definitions of `abstractly unary' for arbitrary binary products (factors through AT LEAST one of the injections) and of `abstractly exclusively unary' (factors through exactly one of the injections) are equivalent by the disjointness part, and are equivalent also to the same notions with binary coproducts of 1 instead of arbitrary binary coproducts (by the universal or stability part). So, *any* of those in this context should mean `connected'. Without those conditions, but with just a terminal object and binary coproducts, then the `at least' part does not reduce to that of coproducts of 1, but the `at most' part does. In that case, the notions correspond to `abstractly unary' and `abstractly exclusively unary', and they are not equivalent. So it all depends on the ambient category.
So far, I have only mentioned BINARY notions of connectedness, but if we want to talk about families of connected COMPONENTS then we must also consider INFINITARY connectedness (as Marta stressed). Here the results for the constructive real line are somewhat surprising.
I still have to digest your disquisitions on constructive analysis, which seem most interesting, but on the above point, I emphasize that inded one must keep the disctinction between connectedness wirt binary coproducts and connectedness wrt arbitrary coproducts (indexed externally, e.g. by a set in a Grothendieck topos E-->SET, or by an object of S in the cae of a bounded topos E-->S. Whether the terminology must make that distinction I am not sure of, or maybe we could say `connected' for the binary case (which is also intrinsic), and `S-connected' for the case of S-indexed coproducts. Once again, under enough hypotheses as I szaid above and was also mentioned by Steve Lack. I am not sure of which hypotheses Vaughan wants to make but, if the minimal possible, then he might need the detailed analysis that I proposed and, in that case, reserve `connected' in the binary case for `abstractly exclusively unary', not simply `abstractly unary', and `connected' when only the binary coproduct considered is 1+1. But if his categories ae categories of graphs, I don't see his problem. With best regards, Marta