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?
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.