Interesting discussion. Technically, the BX has two hydraulic pumps. The Hydrostatic transmission pump and the primary hydraulic pump. Hyper-technically, the oil pump in the engine is a third hydraulic pump. Insano-hyper-technically, there are also two fuel pumps that are technically hydraulic pumps.
Insanity aside, in general tractor lingo, I agree with volfandt. Typically, the HST pump is not referenced as a hydraulic pump when talking about how many pumps a tractor has.
In the case of the BX series, there is one primary hydraulic pump that supplies all hydraulic fluid. This includes the fluid to keep the HST full. The primary hydraulic pump puts out about 5.5 Gal/min unloaded at full speed. The flow diverter maintains about 3.3 gal/min to power steering and HST at all times with the rest going to implements and 3ph. That's why at idle implements won't move. All flow is being diverted to power steering and HST.
The hydrostatic transmission pump and motor are a self contained unit. Meaning the HST pump sends fluid to the HST motor and it loops directly back to the HST pump. The HST pump does not draw fluid directly from the reservoir. Additionally the HST pump and motor do not dump fluid directly back to the reservoir in the transaxle. Small amounts of fluid that "leak" past the piston seals in the HST pump and motor flow back to the reservoir. This is by design. The primary pump supplies fluid to make up for this bypass as well as keeping positive pressure on both the input and output side of the HST pump in either forward or reverse. Without the primary pump, the HST would slowly drain all fluid and stop functioning correctly. The HST can not work without the primary pump.
A one pump tractor almost always has a flow priority valve splitting flow between the power steering/HST loop and the implement loop (if a tractor has Power Steering and HST). Two pump tractors have a pump on a dedicated loop for power steering and HST with the second pump on a separate dedicated loop for implements. The advantage of two pumps is consistent performance of hydraulic implements independent of demand from Power Steering and HST. Additionally, the complex flow priority valve is eliminated. On the down side, more pumps and more plumbing add cost.
As volfandt noted, as equipment gets bigger and functions get added the number of pumps keep going up.
OK, I've rambled enough.