The first through fourth positions are presented to progressing cellists to aid in establishing a mental map of the fingerboard. The development of such a map is an excellent pedagogical goal; however, I have found the positions to be more hindrance than help, both in my own study of cello and in teaching. The problem lies in the nomenclature: there is an underlying irrationality to the numbering convention that mystifies instead of clarifies the fingerboard. Here I explore the cause of the problem, and present an improved alternative nomenclature drawn from my experience with the violin and the guitar.
The first through fourth positions on cello are problematic for two related reasons. The first is the issue of arbitrary-seeming duplicates. This is a stumbling block for many young cellists. Why is there only a single first position, but two second positions– low and high? The second reason is enharmonics. For beginning cellists, the position names on the A string seem to be derived from a diatonic scale — D Major. A reliance on a particular diatonic pattern to understand the numbering convention leads to confusion in keys with flats however, since for example in the key of Bb there is no first position on the A string— it just gets skipped over. Another example of this is the position on the A string where the first finger falls on D# / Eb. The conventional name ‘high third position’ makes sense in a key that ascends diatonically A, B, C#, D, E. However, in the key of Bb this position would more intuitively be called ‘low fourth position’, since the diatonic pitches ascend Bb, C, D, Eb, F.
In contrast to the cello, the violin positions feature a correspondence between finger number in first position and upper position name, and thus the nomenclature is more intuitive. Take again the A string as an example. In first position, because the second finger is responsible for either C or C#, it makes sense that there would therefore be two possible second positions: low and high.
Looking at the cello nomenclature through my guitarist-eyes, I am immediately mystified. Attempting any position nomenclature on guitar beyond a simple chromatic ordering of positions corresponding to frets one through twelve would be irrational.
Taking these observations from violin and guitar and applying them to the cello fingerboard suggests two requirements for a better alternative. First, there should be correspondence between the finger numbers in first position and the higher position names. Second, the basis must be universal chromaticism rather than a particular (and therefore arbitrary) diatonic scale.
The following system meets the above two requirements (A string):
1 on Bb: Half Position
1 on B: First Position
1 on C: Second Position
1 on C#: Third Position
1 on D: Fourth Position
1 on D#: Fifth Position
1 on E: Sixth Position (The intersection position)
Let me know if you have any thoughts or suggestions. I am planning to update all my beginning cello books!
Rock The Cello 
Janos Starkers uses chromatic positions as you describe in his Method Book. But what position it is called is not as relevant as knowing what notes you can play when your hand is in a certain place. In closed position, there is a half-step between each note. In extended position, there is a whole step between first and second finger, and a half step between the others. To make it easier when teaching, I simply tell the students that second position means the first finger plays the second alphabet letter note above the open string (f or f-sharp on d string, for example).
Thanks for your thoughts Tim! I should go back to Starker’s Organized Method, but it always seems like I need a better kind of hand to successfully to play all the exercises, so I usually end up putting the Starker at the bottom of the stack
I like the way you use second alphabet letters, but I find that it gets confusing in the positions where enharmonics are common — when 1 is on D# or Eb on the A string for example. What position is that? Does the name change based on the key? Ack!