"Petroleum based oil" is a general term that usually refers to non-synthetic oils used for lubricating something. An example is motor oil that you put in your car's engine.
"Petroleum distillate" is generally a more flammable product that is use as a solvent. An example is the gasoline you put in your car's gas tank.
Crosman Pellgun oil is basically a high grade of 30 weight, non-detergent motor oil. The key is the "non-detergent" part, because additives that are put in most motor oils to keep the engine clean, will corrode the seals in the gun.
Petroleum - As a technical term, petroleum encompasses the liquid (crude oil), gaseous (natural gas), and viscous or solid (bitumen, asphalt) forms of hydrocarbons that occur in the Earth, but the meaning is often restricted to the liquid oil form.
So the Pellgun oil is a Petroleum based product, because it is a lubricating oil.
The term "Petroleum distillate" could cover a lot of products. Petroleum distillate products include jet fuel, diesel oil, light fuel oil, lubricating oils, motor oils, heavy fuel oils, tars, and asphalt. As well as others that are solvents like mineral spirits, kerosene, white spirits. Most products that come for petroleum are manufactured by a distillation process, which is what the term Distillate means.
Distillate is more commonly used as an abbreviated form of middle distillate.
Middle distillates generally refer to petroleum products in the boiling range beginning at above 160 degC and ending at below 420 deg C -- i.e., between gasoline and heavy fuel oil. These include gas oil, diesel and jet fuel (kerosene).
So if a product has the warning "Contains Petroleum distillate" that is red flag that is should not be used in our air guns.