Where you wrote...
...I think you should say "non-diatonic" notes.
When you select C major as the example it will be true that the notes with accidentals will be "non-natural." But the non-natural aspect is not what is important. The alteration from the diatonic notes of the key signature is what is important.
Let's change the key signature to illustrate the point and use A major.
In A major we have three sharps and the leading tone is G sharp. If we want to lower the leading tone, we alter it by applying a natural! Clearly, being "natural" isn't what makes this note altered. Rather, the accidental is what indicates it's altered.
...in C, would B♭ be just be called flat leading note (or sharp
This may seem like splitting hairs, but...
- "lowered leading" would be a better wording, because we don't want to assume a flat was used for the alteration.
- Also, take care with enharmonic spellings. In C major the submediant is
A. If we "raise" the submediant it would be
A# instead of
My last comment brings up some interesting points regarding functional scale degrees and non-functional chromatic notes. I think it is worthwhile to distinguish between the two (or just skip down to the TL;DR.)
Let's stay in the key or C major and look at the ^6 and ^7 scales degrees.
Over a C major we could have a descending line
C B Bb A Ab G or an ascending line
G G# A A# B C. In both cases we have a tonic chord merely decorated with a chromatic line. In the descending line we have a "subtonic" with the
Bb, in the ascending line that is enharmonically a "raised submediant" with the
A#. Is the
Bb really functioning as a subtonic?
Bb should be doing something vital regarding the key. The subtonic could be introduced as part of modulating to, or tonicizing, the subdominant key/chord. In minor it could be part of the minor
v - when the bass descends by step - or perhaps a modulation to the relative major.
Bb could be used in a borrowed chord context. Basically, if the altered note is acting in a functional way, it should be moving to or hinting toward another key or part of a harmonic resolution.
In the case of a chromatic decoration of the tonic chord we could call the
Bb the subtonic - it's technically correct - but it's not very informative. Maybe a better way to make the distinction is to look at the descending and ascending lines and the alternate enharmonic spellings
A#. If we name them we have "subtonic" and "raised submediant." As mere decoration of a tonic chord is there any meaningful difference?
If we look more closely at the
A# and its potential function, things get interesting.
A# were doing something functional - meaning not mere chromatic decoration - we would want to know what chord
A# belongs to. One possibility is
F# A# C# tonicizing
B. But, in that case the
A# is temporarily the "leading tone."
Although, a "raised submediant" or "raised ^6" scale degree is a common concept when in a minor key. Functionally that has a strong association with the dominant chord where raised ^6 and ^7 are used to form
V7. (Without too much digression, the raised ^6 doesn't occur without the raised ^7 and the raised ^7 is used for the dominant harmony, so I say the raised ^6 is associated with the dominant.)
It's interesting that while a lowered ^2 scale degree is occasionally used in minor there is a common name for that tone. It's usually just called the "lowered supertonic."
All the preceding about function and naming may seem like a confusing mess. In a way that was intentional. Maybe unique names for all chromatic notes isn't necessary. If the terms - tonic, supertonic, mediant, subdominant, dominant, submediant, leading tone and subtonic (along with the special case 'raised' sudmediant in minor) are the standard, why is there a need for additional terms?
Some theoreticians have given additional names, but they are relatively obscure. This source (Anger, A Treatise on Harmony) offers terms like "interdominanat" for a raised subdominant or "hyperdominant" for a raised dominant. Those terms are not standard terms. Depending on the harmonic context, and the function of those tones, terms like "leading tone" of the dominant or "leading tone" of the submediant both use very standard terms and are functionally descriptive.
My suggestion is to use the standard degree names relative to the functional tonic. Non-functional chromatic notes either require no degree labeling for analysis or can be labeled simply by adding "lowered" or "raised" to the standard degree name.