It turns out that while there isn't, in linguistics, a single term to distinguish between two kinds of synonyms, there are at least two sets of terms which discriminate between degrees of synonymy. The exact issue I was concerned with can be found dealt with COGNITIVE SYNONYMY: A GENERAL OVERVIEW.
There we find Cruse:
The scale which [Cruse] has set up consists of absolute synonymy, cognitive
synonymy and near-synonymy.
The scale presented by Cruse is the most general. There also are other
views. Lyons (1981:148) claims that there are absolute synonymy, complete
synonymy, descriptive synonymy and near-synonymy.
What may be of relevance is that I came on this paper earlier tonight while exploring the question, "Is synonym a linguistic term?"
The answer to that, incidentally, would seem to be that while the term "synonym" occasionally crops up in linguistics texts, it is not itself a linguistic term -- synonymy, yes, synonym no. It may be treated as a figure of speech in manuals of rhetoric, but I haven't been able to confirm this.