I can't find any definition for the outputs of Point File Information tool (3D analyst), specifically the pt_spacing field.
Is this the average distance between points, or the average number of points per square foot?
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It is the average distance between points. See here on page 6, 3rd bullet.
Average point spacing
At page 07, also 3rd bullet the document clarifies:
The average point spacing as indicated by the statistics from the Pt_Spacing column. In this example, the average point spacing tends to be approximately 0.6 meter. The lidar dataset used in this paper was captured at a sampling density of two returns per square meter; thus, 0.6 meter gives a good approximation to the ordered capture rate.
Consider the following relationship between point spacing and point density:
I performed a test with a .las file with a known point density of 0.38 pt/ft² (see it here). The expected point spacing result would be approximately 1.62 ft.
In ArcMap 10.1 the result with the Point File Information tool was 1.63 ft.