Blame it on Laitz. I do. He is using the term consonance in two very different contexts and only one of them is right. Consonance is...
consonance and dissonance, in music, the impression of stability and repose (consonance) in relation to the impression of tension or clash (dissonance) experienced by a listener when certain combinations of tones or notes are sounded together. (source - emphasis mine)
Put simply, if you play two keys on the piano at the same time and they sound nice to you, they are consonant. If they make you cringe, they are dissonant. I think the misunderstanding is Laitz's claim that
The weak-beat note must be consonant with both the preceding melody note and the CF.
It's confusing because he's using consonance to describe two successive notes in the first part of his instructions, and to describe to simultaneous notes in the second part.
The weak-beat note must be "consonant" with the note just before it. This has nothing to do with the CF note and everything to do with making your melody something that is reasonably idiomatic to sing. In species counterpoint melodies are supposed to follow certain specific guidelines. Intervals wider than a fifth are not to be used, neither are intervals requiring an accidental (sharp or flat) or intervals forming a "tritone" (augmented fourth or diminished fifth). So in this first idea, he's simply saying that your second melody note should be a second, third, perfect fourth, or perfect fifth away from the first melody note. This is not the standard definition of consonance, nor are these intervals the same as the consonant ones you would use in writing against the CF.
Laitz is saying that the weak-beat note must be consonant with the CF. This is the expected definition of consonance. Your weak-beat note must be a third, perfect fifth, sixth, or octave away from the CF.
For the step/skip/leap business, Aaron is right that
"skips" are thirds. That is, "steps" are seconds; "skips" are thirds; and "leaps" are anything a fourth or larger.
A further misunderstanding here is that many other texts refer to movements of thirds, fourths, or even fifths within a CF note as consonant skips. As far as I know, his use of the term consonant leap to denote a consonant skip of a fourth is unique to him.
From all of this, let's look at the examples you posted.
Measure 6 in 2.15
The CF is on A, the melody moves from C to F. C to F is a perfect fourth, which is a perfectly normal interval to sing and doesn't break any rules. C is consonant with the CF, as is F, so this is completely correct. He could have used E instead for either note of the melody an it would be equally correct. C-E would give melodic motion of a third, E-F of a second.
Your image with F as the CF note
The CF is on F, the melody moves C to E. C to E is a major third, which is also a fine interval to sing. No problem there. But it forms a seventh against the CF, which is a dissonant interval. So this won't work. Here you have two choices. Either move your melody C-D, a major second that forms the intervals 5-6; or move your melody C-F, sounding a perfect fourth and forming the intervals 5-8. Of the two choices, C-D is much better because C-F are both perfect consonances.