costas loop bpsk

I searched some papers and books about BPSK demodulation using Costas loop. I am very confused how it works. I attached all the Costas loops below. They are not consistent though. You can see some one extract the data from the I branch while others from Q branch. Someone give 90 degree phase shift for I while others give it to Q branch.

In Image4.jpg, the hard limit costas loop, at each point, which one is bipolar digital waveform and which one is not? Actually how is the waveform at each branch look like?

Anyone knows the answer? Please help.