No, it doesn't have another name other than BMW coolant.
For now, just focus on bleeding it correctly and ensuring that it is at the correct level. Since you replaced the t-stat before you purchased the car, I'm fairly certainly you did not successfully get all the air out of the system. This would cause the exact symptoms you are describing, and many people cannot get it right on the first try. You most likely have trapped air in the system.
It's most likely not your water pump, and there is no weeping eye on the E36. And contrary to what was said above, the temp needle should never go beyond 12 o'clock. Just bleed your system using the correct procedure and report back afterward.