Very nice. I like the bit of voice leading you put in there and the emphasis on harmonic function. Knowing the names for these things is useless without having some idea of how to put them to work.
I don't know if anyone is interested, but this is the rationale for the naming system:
Dominant - a fifth above the tonic
Mediant - the middle point between the tonic and the dominant
Supertonic - the scale degree directly above the tonic
Then it does a mirror flip. Notice that it's the same thing, but with "sub" as a prefix:
Subdominant - a fifth below the tonic
Submediant - the middle point between the tonic and the subdominant
Subtonic - the scale degree directly below the tonic
Leading tone - describes the harmonic tendency of a subtonic that is a minor second away from the tonic to resolve to the tonic.