Question: What was the first recorded use of someone calling exponential families (in probability/statistics) for which the dimension of the natural parameter space is strictly less than the dimension of the ambient space containing it?
(When the inequality isn't strict, i.e. the affine hull of the natural parameter space is all of the parameter space's ambient space, then the family is called a full-rank exponential family.)
See, e.g. here: http://yaroslavvb.com/papers/notes/lauritzen-expfams.pdf , or Keener, Theoretical Statistics: Topics for a Core Course.