Why do Agda and Coq disagree on strict positivity?
The issue seems to be confusion resulting from a confluence of two factors:


I was using a stale version of Agda (2.3.0.1).  It appears that prior to 2.3.2, Agda simply wasn't checking strict positivity of the indices of constructor results (see the bug I linked elsewhere in the thread).
A closer reading of Dybjer's Inductive Families paper suggests that he may have intended that the inductive type being defined not be bound when typing the indices of a constructor result.  Section 3.2.1 gives the scheme for inductive constructors in prose, and apparently I misread the language describing the binding environments of each portion of the scheme.


This closer reading is of course consistent with the check that Coq and (recent versions of) Agda perform, that prohibit any appearance of T in its own indices.