First, the term pitch class already has a well-defined meaning in music theory, and this isn't it. Basically, pitch class refers to the set of pitches under an octave equivalence relation, generally represented by the integers modulo 12. Typically all enharmonic notations for the same pitch class are regarded as mathematically equivalent, as the general assumption (unless stated otherwise) is 12-tone equal temperament. (And that seems to be the assumption in the question too.)
There's no meaningful sense in which a standard pitch class is greater than another, as the "counter" resets to zero with every octave (unless you want to make some arbitrary definition for a "central" pitch class: C=0 is sometimes used for that purpose).
Anyhow, as to the two "modes" mentioned in the question, Mode 1 is just comparison by pitch (sometimes called pitch height or tone height in psychoacoustics). Aside from the chroma/octave representation in the question (e.g., C4), pitch is sometimes represented mathematically in other ways, such as in the MIDI standard where C0 is simply 0 and each note is given an integer.
As for Mode 2, I'm not quite certain what exactly it is trying to accomplish through mathematical representation. There's no real musical or acoustical sense in which C4 < D♭♭4 in 12-TET. And it's even more confusing to state (as I assume Mode 2 would) that C♯4 < D♭♭4.
The only thing the letter names do in 12-TET is designate the staff position, so Mode 1 and Mode 2 are reflecting completely different elements, one psychoacoustic and the other notational. Why the "less than" relation should even be used in the latter is a little unclear, unless you're writing music notation software and decide to represent staff position numerically. (And I can't think of a common term off the top of my head for such a relation.)
There are musical situations where it makes sense to perhaps compare scale steps in a tonal scale, so even if C♯ and D♭ are played at the same pitch height, they have different tonal function. But in that case, we're usually talking about pitch class (or chroma) where, as mentioned above, we end up in a circular arithmetic space, perhaps modulo 7 when working within a diatonic scale.
I hope this clears some things up, but I'm not aware of a standard term for what you're trying to accomplish with Mode 2 other than informal terms like "letter" or "note letter" or something like that. If all you're trying to do is track staff position or staff height, why not call it that?