Compare over discharge protection disconnect circuits
MOSFETs are voltage activated so they don't need current flow for staying closed/opened. This is good option for our battery powered circuit.
Here are the different options using MOSFETs and what they mean.

Switching the high side (positive power lead) with a N-ch transistor

Vd will be Vbat and Vs=Vout will be close to Vbat (Vbat-Vds) but normally much higher than Vgnd. To open a MOSFET Vgs needs to be more than Vs+Vth=Vout+Vth (where Vth is a threshold that depends on the specific MOSFET). As a result Vgs needs to be higher than Vout, so the load can't hold it open unless it has a second, higher voltage rail it can control. This is a big downside. Reference article.

Switching the low side (negative power lead) with a N-ch transistor

Vd=Vout- and Vs=Vgnd however due to drop Vds, Vout->Vgnd so the load and the disconnect circuit don't have a common ground. In order the open the MOSFET, the uC will need to send voltage that is less than Vs+Vth=Vth, but it can only send Vout>Vgnd. Depending on the load, the MOSFET and the resistors used one can tune the circuit to drop Vout below Vth but it was shown in practice that the result gets too specific for a certain specific circuit and uC. Very easy to get unexpected consequences with the schematic due to lack of common ground. This is a downside. Case study, Discussion

Switching the low side with a P-ch transistor

Opposite to high side and a P-ch. Doesn't make sense. Both disadvantages plus logic is reverse to what we desire.

Switching the high side with a P-ch transistor

We have common ground, so easy for the uC to send a Vgnd signal. P-ch is designed for operating at the high side, so no need for a second voltage rail. The only issue is reverse logic. We want the circuit to be disconnected by default, so the P-ch to be open and its gate at HIGH. When we want to turn on the circuit we need to pull the gate to LOW and we can use a second transistor for this (an N-ch). See circuit #2 and #3 in the question. This is a good option. Reference article, Discussion

Alternative: using an opto-coupler and an N-ch

One way to overcome the lack of common ground and use the N-ch on the low side is to use an opto-coupler. The downside is that it will pass current and consume battery charge when on.