Exactly how the subscripted numbers are given to the different types of boron sheets (borophenes)?
All the subscripted numbers are to denote different phases.
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Distinct phases may be described as different states of matter such as gas, liquid, solid, plasma or Bose-Einstein condensate.
Distinct phases may also exist within a given state of matter. As shown in the following diagram for iron alloys, several phases exist for both the solid and liquid states.
For Borophene, α, β, χ, δ and ψ are different types of structures based on their coordination number(CN) like,

α (CN = 5,6)
β (CN = 4,5,6)
χ (CN = 4,5)
δ (CN = 3,4,5)
ψ (Single valud CN)

Now each different structures can have more than one phases. When any particular type of structure has more than one phases, those are denoted by numbers.
For example, α has 7 different structures. Now the first one is named α. After that total of 6 phases are found. One of which is a very similar structure with α. Hence it was named α'. The difference between α and α' is that there is no corrugation in α whereas α' has a corrugation resulting in better stability than α. However, the other 5 phases are named accordingly from α₁ to α₅ based on their earlier prediction. You can find the different phase structures of α Borophene from the following image - 
Ref:

https://en.wikipedia.org/wiki/Phase_(matter)
Xiaojun Wu, Jun Dai, Yu Zhao, Zhiwen Zhuo, Jinlong Yang, and Xiao Cheng Zeng. Two-Dimensional Boron Monolayer Sheets. ACS Nano 2012, 6, 8, 7443–7453. https://doi.org/10.1021/nn302696v