Is there a name for a cadence which ends on I or i, but with the third intentionally omitted, so as to leave the final chord ambiguous between major and minor? Or a general term for using the tonic chord with the third omitted in a context where a major/minor third would be expected?
In addition to prooffreader's answers, the most common term that I've heard is "open fifth." The idea is that there is a fifth (say, C/G) that lacks the middle third (hence "open").
In analysis, these chords are sometimes identified as
C5, indicating the only chord tone is the fifth above the bass. (This is in contrast to
5/3 figured bass, which suggests both a fifth and a third above the bass.)
Regarding a cadence that uses this, you may consider the general term "attenuated cadence," which simply refers to a cadence that is somehow weakened from its more normative form.
You mean the Quintal harmony, mostly used in Jazz harmony. It is the harmonic structure preferring the perfect fifth, the augmented fifth and the diminished fifth.
Perhaps you find this article more helpful
If it is the end of a three voice harmonic progression it is acceptable for the I to consist entirely of a tripled root. Also, strict counterpoint must end on a perfect consonance (excluding perfect fourths) this naturally results in the chord in question when the last I is just a fifth. I wanted to point out that a I5 does not imply ambiguity between major and minor. You could use it as a common chord for mode mixture to the parallel minor of your starting key. A smoother alternative to this would be using the V as a common chord followed by iv i (plagal progression in minor) this should result in a smooth mode mixture without too much resultant stability thus allowing flirtation with major or with other keys. It could serve as the departure for a developement section.