You may not always get oil mixed with coolant on a blown head gasket. Also most head gaskets never fail. What happens if the engine gets too hot and warps the head and or head plus block causing an imbalanced clamping pressure on said gasket.
I had my ninja do this. Same with a VW I had. Cylinder pressure is extreme and will overcome the gasket and displace coolant.
I would be very confident in saying that it is a possibility but the real issue is why is it brown? It should be in blue. Or any coolant colour that you swapped to. Water alone is bad.
There is much more to this story we are not getting
Bottom line. If that's not coolant, flush it out and fill with proper coolant. Then test for head gasket failure with a proper tester. Very easy, very cheap.
Water alone can over boil that and combine with a bad cap can definitely be your issue.
Start with a 2 cycle flush with distilled water then add coolant and test. You can pressure test the cap as well.
My bet is your cap is bad due to bad coolant.
Start here. Then we will see. Reason I'm saying that is usually the coolant gets displaced during operation. Not after you shut it down.