Yes, you can control how a field is mapped in the spatial join. As you say, the default would be to return a JOIN_COUNT, JOIN_FID etc. and the other fields you have specified in the field mapping.
Of those other fields, you would have the attribute of the first coincident point found, BUT you can control how this is done.
Look at the field map parameter in the tool, and the merge rule dropdown.
from the link I supplied:
Merge rules specified in the Field Map of Join Features parameter only apply to attributes from the join features and when more than one feature is matched to a target feature (when Join_Count > 1). For example, if three features with DEPTH attribute values of 15.5, 2.5, and 3.3 are joined, and a merge rule of Mean is applied, the output field will have a value of 6.1. Null values in join fields are ignored for statistic calculation. For example, 15.5, <null>, and 2.5 will result in 9.0 for Mean and 2 for Count.