Others have already answered the question adequately but I would like to address the issue that seems to be causing a problem: that of the apparent inconsistency in naming between those degrees of the scale beginning with 'sub'.
1) The degrees of the scale have defined names:
Tonic: the 1st degree
Dominant: the 5th degree
Subdominant: the 4th degree (5th degree 'down' the scale, hence 'sub')
Mediant: the 3rd degree
Submediant: the 6th degree (3rd degree 'down' the scale)
Leading note: the 7th degree of the scale
Note that we don't call the 7th degree the 'subtonic' when we consider scale degree for a major key, when there are no accidentals.
2) Chords are built on degrees of the scale and have well-defined 'functions' for instance when used in cadences
Tonic: the 'home key' chord
Dominant: the one that resolves to the tonic
Subdominant: usually resolves to the dominant
Supertonic: Chord ii, resolves to dominant, like the subdominant, so we can say it has 'subdominant function'
Leading note: Chord vii, resolves to the tonic, like the dominant, so we can say it has 'dominant function'
3) 'Subtonic' has a very specific meaning
The term 'subtonic' is used solely to define the note one whole tone below the tonic.
Consider: the term 'subdominant' is telling you to count the same number of steps as the dominant (5) below the tonic. It is something of coincidence that the subdominant is immediately below the dominant, and quite reasonably you might think that the 'sub' just means 'the note below'. When in fact, the prefix 'sub' instructs you which direction to count in from the tonic!
Things become clearer when we consider 'mediant' and 'submediant': the former means count up three degrees, the latter means count down three degrees. (Bonus: the name mediant is derived from being between, or in the 'median position' between tonic and dominant. The same goes for submediant, which is in between the tonic and the subdominant, counting down, of course!).
Now coming to the 'subtonic' and 'supertonic', the same logic applies - count down one step from the tonic in the former case, and count up one in the latter case.
In the Common Practice Era, when considering chord function in a major key, the 7th degree is (almost) always sharpened and therefore always appears as the 'leading note' at cadences. This is not to say that the subtonic will not appear - it will, if there are modulations to the dominant, or if the natural minor scale is used, but in basic music theory exercises in a major key, you probably won't see the subtonic.
I think the key point is that 'sub' in this context does not mean 'a note immediately below', or even more generally 'a note below', rather that it tells you in which direction a note's degree is counted relative to the tonic.
The (excellent) internet source, you mentioned is here, for those who are interested, specifically this page.
See also the Wiki page on Diatonic Function for more information.