Although lead acid batteries are referred to as 12V, they are often at much higher voltages than this during their charge/discharge cycle.
It does seem high though and you may be able to get away with a bit lower, but the nominal fully charged cell voltage is around 2.4V so with 6 cells, you have 14.4V terminal voltage.
To get any current into the cell near a fully charged state you need a supply of greater voltage, hence the 16V suggestion.
Fig 1 in the ti.com pdf is a good picture of the cell behaviour. It shows the sharp rise in cell voltage as the cell is fully charged and you will not be able to see this unless your charging voltage can supply a voltage greater than the 14.4V calculated above.
I'm not an expert in this but did study this area some time ago, so open to other comments and suggestions.