There isn't really an "answer."
Personally, I think I do it differently depending on whether the context is writing or playing.
Writing notation I mentally spell out chords. So, in your example,
C minor, given
i6 and a bass of
Eb, I think to myself "E... C G". Or, I many mental recite up "C E G". It seems I don't think of the sharps and flats when the chord is diatonic. If it's something chromatic then I think of the sharps and flats.
But, when I'm just playing at the keyboard I think much more in terms of intervals. Given an
Eb, I think my right hand will play a perfect fifth or fourth. It's natural to think of letter names while playing, but there is a point to not thinking of letters and thinking of intervals and relative motion. For example, if given a supposed
Eb an told to complete a minor 6/3 chord, you would add
C G, but if asked to complete a major 6/3, you would add
B F# and now your given bass enharmonically changes from
D#. That might not be the a logical explanation, but I seem to tend away from thinking of letters when playing.
Something to think about is how working on textbook harmonization exercises, like your example, you really must think about things vertically, you think about arranging tones above the bass. That makes sense, because the exercise give you a bass part (the RNA is effectively a bass part.)
Compare that to improvising some harmony in a key where you can treat your changes as voice leading. Doing this you can think much more horizontally, linearly. In other words you can treat harmony as voice leading instead of vertical stacking. For example, if you play something like...
...you aren't thinking at the
i6 so much about "what notes should I put above an Eb?" it more about a voice leading pattern, something like "when one a root position triad, hold the fifth, drop the other two voices down by step." I don't care too much what the exact intervals above the bass are, and a really don't care what the letter names are. What I really want is my hands trained to automatically make that move (and other common moves) and my ear trained to know what they sound like.
That scenario doesn't apply to the textbook exercise, but regarding how someone conceives of a 6/3 chord, you can think of them in terms of voice leading and relative motion.