the car is 2004 vw 1.9tdi bora with a pd engine.
it has 90k miles from new on original factory fitted varta battery.
went to start the car,the engine turns over but will not fire.

sounds like the engine turns but a little slower than it should.

glow plug light on dash comes on and off as it should after ii turn the key and before trying to start the engine.

no fault codes showing on vag.com.

battery voltage with tester shows 12.3 volts.

the in-tank fuel pump buzzes normally for a few seconds on turning the key on

( intank pump was replaced a few thousand miles ago).
my impression is partially discharged battery or dying battery.
am i right?
can i charge the battery from a mains charger with battery still hooked up inside the engine bay-(that is without removing the cables?.
any precautions while doing the charging?