Sometimes diminished thirds are respelled to make reading easier, but here's why it can be important to call a diminished third a diminished third.
This post contains three parts: 1) Helping to hear the intervals, 2) some examples of how they arise, and 3) some examples from compositions.
Notice that in all cases (shown here, at least), the diminished third arises from surrounding a target pitch with the notes a half-step above and below. That is, a diminished third follows the "rule" that diminished intervals resolve inward to a unison.
Hearing the difference
Try playing these three exercises. All three, in their penultimate measures, use the same two pitches -- Ab and Bb, enharmonically speaking -- but sound quite different. Repeat the scale measures a few times to orient your ear to the key, then play the ending. (Note: These are not constructed to be realistic examples; just ways to hear the contextual differences in sound.)
[|: EFGA | BAGF :|] [AB]4 | [GB]4|]
[|: ABcd | edcB :|] [^G_B]4 | A4|]
Augmented Unison (Special Bonus Interval)
[|: GABc | dcBA :|] [_A^A]4 | [GB]4|]
Those context-dependent differences in sound are the reason for the distinction of the intervals in theory and when notating.
Diminished thirds don't occur naturally in diatonic scales; they arise from chromatic alterations. For a sufficiently experienced reader, alerting them to the alteration gives information about what's happening in the music and what may be coming.
In LaurencePayne's F7#5 example, the C# indicates a likelihood that pitch will proceed (resolve) to D.
T:F7#5 resolution (d3)
[V:V1] e | d |
[V:V2] ^c | d |
[V:V3] A | B |
[V:V4] "_F7#5"F | "_Bb"F |
Respelling the chord to use Db suggests a different resolution (and renames the chord). (Note: I've displaced the b13 by an octave for the illustration).
T:F7b13 resolution (M2)
[V:V1] e | d |
[V:V2] _d | c/2 B/2 |
[V:V3] A | A/2 B/2 |
[V:V4] "_F7b13"F | "_Bb"F |
Another respelling (and revoicing) of the chord gives an inverted augmented sixth chord and, again, another resolution.
T:Faug6 resolution (d3)
[V:V1] a | a | ^g | a |
[V:V2] f | e | e | e |
[V:V3] ^d | e | d | c |
[V:V4] "_Faug6"c | "_Amin"c | "_E7"B | "_Amin"A |
In the wild
A melodic diminished third is found in Schubert's “Der Müller und der Bach” from Die Schöne Müllerin (D795/19), measures 8-9.1
Bach's Fugue in D minor (BWV 851) from WTC I contains a diminished third in m. 12.3
Fugue in D minor m. 124
In Brahms's German Requiem, a diminished third (one of many diminished and augmented intervals in the piece) can be found in the Alto part at m. 72.5
Ein Deutsches Requiem m. 72, vocal parts6
Note that were Brahms resolving to Gb Major rather than F major, he would have written the Alto B as a Cb; i.e., a minor third with the tenor.
1Adapted from https://milnepublishing.geneseo.edu/fundamentals-function-form/chapter/31-the-neapolitan-chord/
2Mandyczewski edition from IMSLP
3Adapted from https://www.teoria.com/en/articles/2017/BWV851/01.php
4J.S. Bach, Das Wohltemperierte Klavier, Teil I Urtext (1997, Henle), p. 28.
5Adapted from Ouida Clemons, "TYPICAL ELEMENTS OF BRAHMS'S CHORAL STYLE AS FOUND IN THE GERMAN REQUIEM", M.A. Thesis, North Texas State Teachers College (1942). See PDF page 52.
6Mandyczewski edition from IMSLP.