Why is there disagreement on the number of surfaces of the human heart?
Heart surfaces are not clearly anatomically defined but arbitrary determined in the context of the heart orientation, that is the relation of the heart to other organs. Different sources mention 3 or 5 heart surfaces and there seems to be no "official" agreement about this.

Most sources (Dartmouth College, Gray's Anatomy for Students 2004...) agree about at least 3 surfaces:


Anterior (or sternocostal)
Inferior (or diaphragmatic)
Left pulmonary


Many sources (TeachMeAnatomy, KenHub...) mention additional 2 surfaces:


Posterior (or base)
Right pulmonary


The reasons for disagreement:


The heart base may not be considered a distinct surface (as in the picture).
The right pulmonary and anterior (sternocostal) surface are often considered one surface.




Picture: Heart surfaces (source: Echo Class 101)