The charger mode and boost mode are two totally different functions done a different times.
Charge mode is putting charge into the battery and maybe also running a system load.
Boost mode is that you want to remove the adapter and plug in a printer or LED light or something to power from the battery. In this mode the BATFET is on making the BAT pin and SYS pin the same voltage minus the IR drop. The System voltage is appled to the inductor and the converters FETs operate as a boost charger suppling a 5V output on the VBUS line. The system may still be running, but it is off of the battery power.
More information about the charging mode and difference between the SYS output and the BAT output.
One way to look at is is the charger is a power path part that supplies at least 3.65V to the system (called Vinsys=3.5V).
The system output has a LDO charger on it to charge the battery.
IF the battery is below the min-sys-votlage, say 3.3V, then the BATFET adjusts its resistance to allow the Program charge charge current to the battery while maintaining the minimum system voltage....so if Vsys = 3.65V, Vbat = 3.3V, Ichg = 1A then the BAT FET resistance was adjusted to 0.35V/1A = 0.35 ohms.
As the battery charges to the min_sys threshold the BATFET turns fully on minimiIng the IR drop and causing a slight IR drop at the system voltage which is why the system voltage regulated higher initially so after the BATFET fully turns on the system voltage remains above 3.5V.
At this point the BATFET is fully on and the charge current is regulated off theRDSon resistance of the BATFET.
Another way to say it is that the system voltage will track the battery votlage when above the min_sys_voltage and isolate itself from the battey votlage is the battery votlage is below Vsysmin.