"3/8"-16 into the head, 3/8"-24 under the nut.
There are good engineering reasons for doing so, but there's not enough strain to apply in this case. Replace with 3/8"-16 full length, and call it good. That's what I do when I have to replace a stud - usually cut a 2" or so length of 3/8"-16 brass rod and LocTite (#242) the thing in place Replace the nut with a 3/8"-16 UNC piece, and reuse the dished washer (it's important.)"