It's the part the oil filter housing is attached to, the heat exchanger part is directly underneath, a square metal box.
Looks like this
https://www.ebay.co.uk/itm/393099168586
Coolant flows around the exchanger cooling the engine oil.
By the way you describe the sound and where it comes from, it's in that area.
I wonder if the oil filter and pump is draining back to the sump and it's taking a few minutes of running to prime itself again, that would explain the noise and the oil pressure light.
I presume there's the right level of oil, so the pickup pipe should be fully submerged.
By the pictures of your old oil filter, there's something not right there and I would think whatever valve stops the oil draining out of the filter back past the pump would be in that filter/heat exchanger housing??