I don’t know if you can visually tell an AGM battery from and Lead acid battery but the original battery looked like a typical lead acid battery.
It probably said AGM or VRLA on it if it was AGM.
If it were me i would put a quality new AGM in it. Not necessarily an OEM one.
According to Varta they appear to only recommend AGM for a 2008 Cooper S.
https://www.varta-automotive.com/en...(R56)","engine":"Cooper+S+(120+kW+/+163+PS)"}