It sounds like there's not geographic (datum) transformation set in the map, either when you check it now, nor when the service was created. Around 10.3, we started adding a file called gtdefaults.json to ArcGIS Server. If there's no transformation set, it will look for one in the file. Look in the pedata folder of the Server install. (or in Desktop's install).
ArcMap doesn't use the file, even though it's available. Pro does.
The line in the file is:
[ 4269, 4326, 108190, 1, 0, 0 ],
which translates into NAD83, WGS84, WGS_1984_(ITRF00)_To_NAD_1983, "apply in opposite direction"
If you want to change this or just test it, edit the file, and replace 108190 with a zero. You may need to restart the service to pick up the change.
Melita