Hi Barkuti,
Yes, the circuit that you have designed is the full complement of the original circuit, so it will have the same characteristics as the original circuit, except all polarities will be reversed, as you show.
About MOSFET gate threshold voltage (VGth): The VGth must be low enough so that the MOSFET turns on fully with a low battery terminal voltage. That is the criteria, rather than back flow. There are many low VGth MOSFETs available, both N and P types. The FDS6575 PMOSFET, in the original schematic, has a VGth of 400mV to 1V5 and turns on to a low drain/source resistance (Rds) of 17mR at 2V5.
https://www.onsemi.com/pub/Collateral/FDS6575-D.pdf