When Linda McKnight tells me to do something, I do it and think about it later. And since the system works so well, I just do it without analyzing why. Therefore, my explanation is my reading of why it works, not hers. And, I was taught to explain upshifts in terms of where 1 goes, and downshifts in terms of where 4 goes.
Re terminology: Assume downshift on G string
from 1-C, 2-Db, 4-D
to 1-Ab, 2-A, 4-Bb
In this case 4 goes to Bb, which is 1 whole step below where 1 was (C). I was taught to think "4 goes 1 under 1".
from 1-Eb, 2-E, 4-F
to 1-D, 2-Eb, 4-E is described as "4 goes to 2" (from F to E)
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from 1-Eb, 2-E, 4-F
to 1-C, 2-Db, 4-D is described as "4 goes half under 1", i.e., D (4) is a half step under Eb (1)
So in every downshift, I must say where 4 goes in comparison to the prior position. All the above applies no matter what finger is being bowed in either position.
To repeat: initially, this is slow and confusing, and is best started with a teacher explaining and watching. Working your way through a Marcello sonata movement can take 4-5 times as long. But sooner than you expect, the fog lifts and you can analyze the shift on the fly, in an instant. Also to repeat: When McKnight hears a shift slightly off, she says "explain the shift", and I do, and when I play it again, intonation self-corrects. Every time. Finally, I see it as one more way to take total command of the instrument.