Our BLS implementation returns only one type of bls "power" = the square of the "signal residue" from https://arxiv.org/abs/astro-ph/0206099.
The astropy BLS implementations give additional conventions for the "power": https://docs.astropy.org/en/stable/timeseries/bls.html. We should (1) be very clear about what exactly we are calculating and returning so that we avoid confusion, and (2) if possible, allow user to specify which convention they want to have returned.
I believe the same is also true for Lomb Scargle -- we're only returning a single convention. In LS we are clear in the documentation about what convention we are using but it might be good to allow alternatives.