I don't think we can tell whether it's the original, but it very well could be. If it's not original, it is obviously a genuine replacement. If it happens to be original, it's now 11 years old, which would be very good going. There is a good chance that it's starting to fail and another winter would probably see it off.
The faults you're seeing could well be caused by low voltage when cranking. That's often a sign of a battery starting to go bad as cell resistance increases as lead sulphate coats the plates. At high current draw, even small additional resistance results in appreciable voltage drop. These sort of messages tend to crop up before any sign of failing to start.
I would also make sure you're confident in the alternator as failure to fully charge the battery can give similar symptoms and under charging would damage your new battery if you were to replace.
Alternators tend to fail with bad diodes or ineffective voltage regulators. For these symptoms, voltage regulator most likely. Bad diodes tend to lead to loss of charging current or battery drain overnight. The voltage regulator sets the alternator output voltage to avoid damaging electronics or overcharging the battery.
Voltage regulator is built into the alternator and is crucial. If it shuts off too early, battery won't fully charge and this can lead to sulphation of the plates. You really need an accurate digital voltmeter to test at the battery posts. With engine running (idling) for a few minutes and minimal electrical load, voltage at the battery posts should be somewhere between 14.2v - 14.4v. If less than that, test at the alternator output, which should be about 14.5v. Current: 2010 Range Rover L322
Vogue SE, TDV8 3.6, Bournville
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Now gone: 2002 Range Rover L322
Vogue, TD6, Giverney Green
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