As others have mentioned, the word diatonic comes from ancient Greek music theory and literally means "through [whole] tones." Ancient Greek music tuned its scales using intervals of perfect fourths called tetrachords. A diatonic tetrachord was one that was tuned with two whole tones on the top, and the remainder left on the bottom (roughly a semitone), like descending E-D-C-B in our modern scale.
These tunings in ancient Greece were contrasted with chromatic methods of tuning the tetrachord, which generally involved intervals smaller than whole tones and therefore often resulted in some consecutive semitone-sized intervals, like we find in our modern chromatic scale.
The reason this background is important is because it gave birth to two somewhat different ways of using the term diatonic today:
(1) The first comes directly from Greek scale construction. By adding an additional diatonic tetrachord (A-G-F-E) to the E-D-C-B one I mentioned above, we can get a complete descending scale for an octave: E-D-C-B-A-G-F-E, with a distinctive pattern of whole steps and half steps. (I give the scale in descending order, as that was typically how Greeks would think of tuning it.)
Those notes were also the "white notes" on our modern piano. This pattern derived from the Greek diatonic scale thus gave birth to all the diatonic medieval modes, from our modern C major and A natural minor scales, to the Dorian, Phrygian, Lydian, and Mixolydian modes, all employing that same diatonic scale from ancient Greece. In that original sense of diatonic, all the modes using that specific set of notes with its pattern of whole steps and half steps (like the white keys on a piano) are derived from a "diatonic scale."
(2) The second usage comes later historically, not emerging until around the 18th century when our modern major/minor key system came into place. During the late 16th century, there was a strong revival of interest in ancient Greek music theory that began in Italy and spread more widely. With it, the concept of things like chromatic tunings became of more interest. Chromatic notes became associated specifically with notes outside of the standard diatonic scale (consisting, again, mostly of what we think of as the "white notes" on the piano).
Thus, when major and minor keys became standard in the 18th century, the word diatonic retained its association with the "primary notes of the scale," while chromatic referred to notes "outside the primary scale." For major keys, the seven standard notes were thus diatonic, and any accidentals could be considered chromatic notes. For minor keys in classical style, it was a bit more complicated, as the 6th and 7th scale degrees often were employed in both raised and lowered forms. Traditionally, the word diatonic related most directly to the so-called natural minor, because of the rationale from point (1) above.
However, modern music theory sources are inconsistent with the way they use diatonic in relationship to minor keys. Generally speaking, standard uses of both the raised and lowered forms of scale degrees 6 and 7 in minor are often discussed as "diatonic" in many modern music theory books. That is, the leading tone is generally raised in dominant function chords. The sixth scale degree can be raised to progress melodically in smooth motion to the leading tone. The sixth scale degree is generally flattened in other contexts (especially when leading to scale degree five) and the seventh sale degree may also be flattened to progress melodically to the lowered sixth. All of these are often thought of as standard "diatonic" patterns in minor, while more exotic uses of chromaticism with scale degrees 6 or 7, or uses of other accidentals on other scale degrees, would be considered "chromatic."
This sidesteps the question of whether melodic and harmonic minor scales are diatonic, which is really a matter of opinion and how exactly you create the formal definitions.
From my perspective, the usefulness of the term diatonic in the second sense is not about scales specifically, but rather about whether particular notes, intervals, and chords are considered "chromatic" or not. To my mind, if the goal of the word diatonic now is to determine the standard uses of notes of the scale, then a leading tone in minor is necessary in classical style, so it's obviously "diatonic" by the second meaning. Intervals and chords built using that leading tone are also, by extension, "diatonic." Hence, the part in the OP's original question that asks whether i-V-i is "diatonic" is that such a progression obviously is, according to the second meaning.
But is a harmonic minor scale "diatonic" though? I'd personally say no. My rationale is that generally when people speak of "diatonic" in relation to scales, they are using that historical definition (1), with its specific pattern of whole steps and half steps. Some would allow other scales to be labeled "diatonic" too, but the criteria get increasingly loose if we were to admit harmonic minor (with its augmented second) into the mix.
On the other hand, chords and harmonic progressions that use the standard minor key accidentals are thought of as "diatonic" according to definition (2). By that definition, we might consider a use of a raised 6th scale degree fine (and "diatonic") if necessitated by melodic motion, but a purely coloristic use of the raised 6th degree (as in a simple i-IV-i) progression might be thought of as a "chromatic" variant of the iv chord.
I think the usage I outlined in the last couple paragraphs accords with how standard music theory texts often use the terms today, but there is still some variation in official definitions and use.