Cutting the rhizome before dividing?

[Ray]

Such sympodial plants live as "colonies" of individuals, with each one contributing - via photosynthesis, water- and nutrient uptake, or simply as storage vessels - to the overall well-being of the colony.

I suspect the partial cuts are interpreted as damage to the integrity of that colony, so the response is to "reactivate" the dormant "eyes" around the damage for reinforcement. A complete cut, on the other hand, indicates the complete loss of that section, making the colony smaller, so the need for those resurrected resources is reduced. For the severed section, whether or not there is new growth is dependent upon the size, age, and condition - i.e., is the resource supply sufficient to support new growth that will ultimately save and support the new colony.

[Fairorchids]

When you cut Catts, you should have min 3 (and preferably 4) mature bulbs in the lead division.

Unless you have 'a lot of plant' to work with, the backbulbs should generally be left alone - unless they have a lot of live roots, in which case you could cut them up into 2-3 bulb sections.

Story of a single bulb division (Guarianthe skinneri 'Debbie' FCC/AOS):

I was given a single bulb, which had fallen off a mature plant.
First year it gave me a single bulb, about 1" tall.
Second year it gave me a single bulb, about 4" tall.
Third year it set two new growths, each about 7" tall.
This summer it is growing 3 new growths, which will probably end up just short of blooming size.
So, from that single back-bulb, it will be 5 years to a BS (but multi-lead) plant.