Mk5 2.0 TDI not starting - Oil pressure warning lamp - what to do?

[RigoX]

Finally some updates: Car is starting again. Jump started it with an external charger to get the engine going.
Looks like it was indeed to weak (while somehow still holding plenty charge). Car starting without problems for now.
But looks like I'll have to replace the battery.

I attached a picture of the battery.
Is replacing the battery a DIY job? If so, where and what to buy as a good replacement?
What specifications to look for?
BTW, based on the picture - is this still the OEM battery?

For warranty issues (not just for the battery itself), is it better to get it professionally replaced?

Thanks again for all your help!