Why is a laptop power supply output voltage different from its battery?
The voltage on your battery "10.8V" is the "nameplate" voltage, some average voltage that your battery delivers over full discharge cycle. The value of "10.8" indicates that this is a battery of 3 Li-Ion cells in series, giving their standard "nameplate" voltage of 3.6V per cell.

Charging the Li-Ion cells requires variable voltage levels, from 2.5-3 V per cell (when in deeply discharged state) to 4.2V (4.35 in some cases) per cell in "constant-voltage" stage of charging process (otherwise the cell won't be charged to full capacity). So the feeding power must have some overhead to provide the charging process (or let internal charger to do so). So, for 3-cell, it comes up to 12.6 - 12.9 V of input. The external power supply must provide this headroom, which includes minimum of "drop-out" (or regulation) voltage for switching electronics inside the external power supply and internal charger, 1-2 V per device, give or take. Eventually it comes up to 12.9+4 = ~ 16-17V.

The "19.2" nameplate is a bit of mystery, since it is not that stable in the first place. It is just an industry standard. Any AC-DC adapter in the range of 18 - 22 V will happily charge your laptop, very likely.