Halliday & Matthiessen (2014: 642):
But the most important distinction, which again follows from the different nature of the two types of relationship, is that in ellipsis the typical meaning is not one of co-reference. There is always some significant difference between the second instance and the first (between presuming item and presumed). If we want to refer to the same thing, we use reference; if we want to refer to a different thing, we use ellipsis: Where’s your hat?—I can’t find it.—Take this (one). Each can take on the other meaning, but only by making it explicit: another hat (reference, but different), the same one (substitution, but not different). Thus reference signals ‘the same member’ (unless marked as different by the use of comparison); ellipsis signals ‘another member of the same class’ (unless marked as identical by same, etc.). The difference is most clear-cut in the nominal group, since nouns, especially count nouns, tend to have clearly defined referents; it is much less clear-cut in the verbal group or the clause.
Within the nominal group, ‘another member’ means a new modification of the Thing; Deictic (this one, another one, mine), Numerative (three, the first (one)), or Epithet (the biggest (one), a big one). In the verbal group, it means a new specification of polarity, tense or modality through the Finite element (did, might (do), hasn’t (done)); and there is a slight tendency for ellipsis to be associated with change of polarity and substitution with change of modality. This tendency is more clearly marked with the clause, where ellipsis adds certainty (yes or no, or a missing identity), whereas substitution adds uncertainty (if, maybe, or someone said so); this is why, in a clause where everything is ellipsed except the modality, it is quite usual to use a substitute (possibly so, perhaps so) unless the modality is one of certainty – here we say certainly (elliptical), rather than certainly so:
Have you got a nicorette on you? – Certainly.
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Reference: Steve played badly and he was soon eliminated.
Ellipsis: Steve played badly and Ø was soon eliminated.
That is, in these typical uses of reference and ellipsis, the relation in both is one of "co-reference".