You should specify a minimum relative compaction value for pipe trench and general earthwork. Note that the dry density of the soil will be embedded in that relative compaction value (dry density as compacted divided by maximum dry density obtained during testing).
Dry density is used because it is esentially a "normalized" density value. That is, the moisture content and saturation variables are removed from the dry density value. How you arrive at that dry density (and percent compaction) does depend on moisture content however, which is where some specification for moisture limits should be considered based on your Proctor testing. Review the Proctor test for additional clarification on that.
When you see wet density used or reported in conjunction with nuclear gauge testing, it is shown probably for your information and is really sort of an intermediate step in arriving at the dry density value and ultimately, the percent compaction (or sometimes relative density) value that you've specified.