What is the common way to name modes of, say, the harmonic minor scale? Can one say "D harmonic minor Dorian" and be understood to mean the second mode?
"D harmonic minor dorian" does not make sense, because "dorian" doesn't mean just a second mode of some scale, but it has a very concrete meaning, namely a scale with a minor third and a major sixth.
The names for the modes of harmonic minor are not as standardized as the names for the modes of the major scale, but you would often see the following names:
I. Harmonic minor II. Locrian #6 III. Ionian #5 IV. Dorian #4 V. Phrygian Dominant (= phrygian #3) VI. Lydian #2 VII. Mixolydian #1 (Ultralocrian, Altered Dominant bb7)
As you can see, most names are derived from names of major modes, e.g., dorian #4 simply is a dorian scale with a raised 4. The name of the seventh mode appears to be the least standardized. Following the logic of the other names, it should be mixolydian #1, but most people tend to not recognize the relationship with mixolydian because in this case it is the root which is raised. I've also seen superlocrian for the seventh mode of harmonic minor, but this name is problematic, because it is also used for the seventh mode of melodic minor (among others).
For the names of the modes of melodic minor you have even more theoretical possibilities because the melodic minor scale can be viewed either as a major scale with a flatted third, or as a dorian scale with a raised seventh:
I. Melodic minor II. Phrygian #6 | Dorian b2 III. Lydian #5 = Lydian Augmented | [Phrygian b1; not used] IV. Mixolydian #4 (#11) | Lydian b7 = Lydian Dominant V. [Aeolian #3; not used] | Mixolydian b6 VI. Locrian #2 (♮2) | Aeolian b5 VII.[Ionian #1; not used] | [Locrian b4; not used] Altered Scale, Superlocrian
As indicated above, some theoretically correct names are simply not used, especially the ones where the 1 (root) is raised or lowered. Furthermore, the seventh mode has a special name because, at least in Jazz, it is used frequently. Its most common name is "Altered Scale" because it contains all alterations over a dominant seventh chord (#5=b13, b9, #9 #11=b5).
The most unambiguous way to refer to these modes is a little verbose, but is common in the academic world when discussing rock/pop/hip-hop music especially. You just number the modes, and then say "xth mode of harmonic minor on tonic" or "yth mode of melodic minor on tonic." Alternatively, you could say "nth harmonic minor mode on tonic. If a piece had G as tonic and followed the standard harmonic minor scale, we just say "G harmonic minor" (not "first mode of harmonic minor on G"), but if a piece has G as tonic but the notes of C harmonic minor, we would say "fifth harmonic minor mode on G." However, this particular mode is so commonly known as the Phrygian dominant, we often just use that. The longer locution is more necessary for less common harmonic minor modes and for the melodic minor modes.
Again, it's an admittedly awkward phrasing, but it does away with ambiguity and less-agreed-upon naming conventions. There are also some legacy terms that most now consider offensive (such as anything involving the word "gypsy"), which this system avoids as well.
I believe the different mode names are only relevant to major scales. In a harmonic scale(a minor), it doesn't work. Instead, you change the degree of the scale to suit your harmony by starting at whatever note you want to suit what sound you wish to achieve. Your harmony is there to complete the melody and is not as "rigid" as a major scales. Scales you learn are a basis for explorations. You can use them in any way you want. There are dozens and dozens of them, all derived from a basic major. Hope that helps Bill