Quote "The distance between them is the same, but the first one is about 5/8 inch from the receiver, the second one is 3 inches."

Makes me think the barrel was set back about 2 inches. That would have take the usual factory marks out. If done correctly it could be safe and shoot good. You will not be able to use a original hand guard with that barrel, might be a candidate for a sporter type stock and make a deer rifle out of it.
The smooth Krag action will make a sporter that you will really enjoy if you plan on keeping it and passing it down in the family, the only down side is it will not ever sell for the money that you put into it.

EDIT
Just for meanness check and see if you have a plugged hole for the rear sight. I just checked an old barrel and the close screw hole was where you described but it was added by a gunsmith or Bubba to move the sight back. The original forward hole is plugged on mine. Even if you have a forward hole plugged by an expert there should be tell tale signs.