Stove oil? As in denatured alcohol? Hmm not sure I have heard that one. Going for a google search on stove oil.
Edits and interesting tiddy bits;
Off road Diesel #2 and Home Heating Oil / Fuel Oil # 2 apparently differ only in the tax applied at the time of sale.
Both are dyed red. They differ from standard Road Diesel #2 in that road tax has been applied, no red dye has been added, and there is a specification regarding ash and/or sulphur content. In extremely cold climates, Standard Road Diesel #2 may be blended with Diesel #1 or Kerosene K-1 to assist with gelling problems resulting from the cold weather. Similar blending may occur on home heating oil / fuel oil based on your geographic region, anticipated temperatures, and overall climate.
Home Heating Oil / Fuel Oil #4 and Home Heating Oil / Fuel Oil #6 have a higher BTU content. This is normally accomplished by adding much heavier petroleum oil distillate similar to automotive motor oil to the formulation. This allows a furnace to burn less fuel, for each BTU of heat produced, as well as often times reducing the cost per gallon of fuel. These types of Home Heating Oil / Fuel Oil should NOT be used in a Diesel Fueled Engine or Generator.
... not exactly sure what you mean by "stove oil" or "furnace oil", but it sounds like what we call kerosene & diesel (or fuel oil) here.
Back when I lived in a mobile home I burned kerosene in the oil burner. It was because the tank was outside and regular fuel oil would jell in the cold weather, and the kerosene wouldn't. That may be why you were told to burn the lighter grade oil. Kerosene may be more expensive, but a tank full of the jelled cheaper fuel oil isn't worth anything when it's cold and the burner won't run.
Hmmmm I was told buy my distributor, #2 fuel IS ULSD cut with kerosine to ensure cold weather performance........stove oil.
Oh, there is a lot of keyboard noise made over this topic..... Do a search on #2 fuel or ULSD.
Main difference is price due to different taxable uses and such. Sometimes die color varies also adding to the questions