Refer to figure.
The pressure pump in a piston aircraft engine lubrication system must deliver oil reliably under high and consistent pressure to all bearing surfaces throughout the full range of engine operating conditions. Several pump types exist in engineering, each with different characteristics of pressure capability, flow consistency, self-priming ability, and tolerance to contamination. 

The gear pump is universally adopted as the standard oil pressure pump in piston aircraft engines because of its specific combination of advantages: it produces a consistent positive-displacement flow at high pressure, it is mechanically simple with very few moving parts, it is highly reliable and durable, it is self-priming, and it tolerates the viscosity variations of engine oil across the full temperature range from cold start to operating temperature.

The gear pump consists of two meshing spur gears rotating within a close-tolerance housing. Oil is carried in the spaces between gear teeth around the outside of both gears from the inlet to the outlet, where the meshing of the teeth displaces the oil under pressure into the delivery line. The close clearance between the gear teeth and the housing walls prevents backflow and maintains pressure.

Diaphragm type pump → INCORRECT. Diaphragm pumps use a flexible membrane that oscillates to draw and expel fluid through check valves. They are used in aircraft fuel systems (notably as engine-driven fuel pumps on some carburetted engines) and in vacuum systems, but they are not used as oil pressure pumps in piston engine lubrication systems. 

Gear type pump → CORRECT. The gear pump is the standard and universally used oil pressure pump in piston aircraft engines. Its positive-displacement design generates consistent high pressure regardless of engine speed variations. Its simplicity, reliability, self-priming capability, and compatibility with engine oil viscosity characteristics make it the ideal choice for this application. Both wet and dry sump systems use gear pumps as their primary pressure pump element.

Scavenge pump → INCORRECT. The scavenge pump is a component of dry sump lubrication systems and its function is to return oil from the engine crankcase to the external oil tank after it has lubricated the engine components. It operates at low pressure on the return (scavenge) side of the circuit. The scavenge pump is typically also a gear type pump but it is not the pressure pump that delivers oil to the engine components. 

Vane type pump → INCORRECT. Vane pumps use a rotor with spring-loaded sliding vanes that sweep oil from inlet to outlet within an eccentric housing. They are used in some hydraulic systems and certain fuel boost pump applications. However they are not the standard choice for piston engine oil pressure pumps due to their greater complexity and the potential for vane wear at high pressures and with the viscous nature of engine oil.