Why does my sauce split?
To address the part where you asked for the difference between your mother's sauce and yours: yes, the kind of peanut butter may well be responsible for the difference. The Internet gives the ingredients as 


  ROASTED PEANUTS, SUGAR, HYDROGENATED VEGETABLE OILS (COTTONSEED, SOYBEAN AND RAPESEED) TO PREVENT SEPARATION, SALT 


or, for a low-fat version 


  INGREDIENTS: ROASTED PEANUTS, CORN SYRUP SOLIDS, SUGAR, SOY PROTEIN, SALT, HYDROGENATED VEGETABLE OILS (COTTONSEED, SOYBEAN AND RAPESEED) TO PREVENT SEPARATION, MONO AND DIGLYCERIDES, MINERALS (MAGNESIUM OXIDE, ZINC OXIDE, FERRIC ORTHOPHOSPHATE, COPPER SULFATE), VITAMINS (NIACINAMIDE, PYRIDOXIDE HYDROCHLORIDE, FOLIC ADIC).


Your mother is using a product which has been specifically engineered to prevent separation. Especially the low-fat version also includes mono- and diglycerides, which are also a good emulsifier. You are using some product whose selling point is probably the low degree of processing. I suspect that it is not only the ingredient difference, but also the microstructure of the fat particles that play a role (the two may have been milled differently). 

So, one option would be to use Skippy too. It would likely get you the result your mother gets.