If you fall under IRC, IRC requires most framed walls to include a Class I or Class II vapor retarder on the warm side of the wall during the heating season. For NJ, that would be the interior side. Vapor retarders are separated into three classes: Class I: Less than or equal to 0.1 perm (6-mil polyethylene); Class II: Greater than 0.1 perm but less than or equal to 1.0 perm (Kraft facing); Class III: Greater than 1.0 perm but less than or equal to 10 perm (interior latex paint on drywall). There are exceptions: if the wall has an adequate thickness of exterior foam sheathing OR if the wall is vented, ie, it has an exterior rain screen between the sheathing and the siding, then you can use a Class III retarder, which can be simple latex paint on the interior walls. If you use kraft faced insulation, then staple the paper to the 1-1/2" wide edge of the stud. Do not staple it to the side of the stud inside the framing bay. As to moisture movement, yes, 6-mil poly will stop diffusion and drying whereas kraft paper allows some albeit at a slow rate. If you tend to have exterior siding (like vinyl siding) that easily passes water, and if you don't have any housewrap under that vinyl siding to shed moisture that gets behind the vinyl siding, then Kraft paper may be the better choice. Walls can get wet, then when the sun hits the wall, moisture can be driven into the wall assembly from exterior to interior. If moisture gets into the wall, you want it to be able to dry. 6-mil poly really only allows drying back out to the exterior, reversing the path that allowed it in. Kraft paper, with it's higher perm rating, allows allows a bit of drying to the interior. So you are correct, 6-mil poly is indeed a better vapor retarder than kraft-faced insulation with regards to stopping vapor diffusion. But you might not want "the best" vapor barrier in your wall assembly if you have lousy walls. The whole vapor barrier/retarder business can be quite confusing. Most inspectors don't even understand it. The only time I usually recommend slashing the kraft facing on FG batts is when someone is adding more kraft-faced insulation over existing kraft faced insulation, or if they are adding 6-mil poly over the entire wall after kraft-faced batts were already installed. Remember, paint is quite permeable. Grout is permeable. So you would not want to slash the kraft facing behind painted or tiled drywall.

Building paper is not a vapor retarder. It is often used on the outer face of a building because it is vapor permeable. There are synthetic roofing underlayments that are vapor retarders but I can't think of any reason to use them on a wall. Kraft paper facing on fiberglass insulation acts as a vapor retarder when the edges are well stapled to studs but it usually has an asphalt coating on the insulation side so you probably shouldn't be using it. A better vapor retarder is foil-faced insulation and the best is a 6 mil polyethylene sheet.

"Mostly heating, but the house does also have central air." How many degree days of heating? degree days of cooling? PA is all over the place. Typical winter low? Summer high & humidity? All these things come into play. In many parts of the the US a vapor barrier is not actually doing anything much. It can only be on one side of the insulation, and with relatively equal heating and cooling loads should just be omitted. Frost freezing in cold attics is a very cold weather occurrence. It simply does not happen in the more temperate areas. One of the problems we have is trying to establish energy codes that cover insulation and things like vapor barriers that are uniformly applicable across the whole nation. With the wide range of weather it is simply destined to fail.

I have to agree with mightyanvil, I can't think of any purpose to stapling building paper to the inside of the siding. If you have enough rain water infiltration to create "puddles" you have more serious problems. If it's an occasional errant leak onto the inside of the siding, I don't see that harming the fiberglass in any meaningful way. Now if you were putting in loose cellulous (which is more succeptible to moisture damage) you might have a point. I would guess the mold you see is more likely from condensation forming on the siding from warm inside air hitting it. Make sure your inside air barrier (i.e., the drywall) is as tight as possible and perhaps leave an inch of airspace between the siding and your batts (if possible) Another way to keep moist inside air from hitting you siding would be to cut styrofoam to be slightly smaller than each stud bay, then apply spray foam (e.g. great stuff) into the cracks all around to hold it in place. Even with this method, I would leave a little air space between the styrofoam and the siding to prevent any moisture from being trapped against the siding. And of course plug up any wire and pipe holes with caulk or great stuff as well.