In the context of fuels, there are two main uses for the term "distillate". For both, it's an abbreviation of "petroleum distillate". Petroleum get used for fuel in 4 different forms:

(1) Directly, as crude oil. No refinery needed! Light crudes, with a larger proportion of lighter (i.e., short-chain, lower molecular weight) hydrocarbons are usually better for this.

Otherwise, the crude oil goes into the refinery, where it is distilled. The distillation process involves boiling off and collecting and condensing the lighter hydrocarbons. The lightest ones that boil off first end up as petrol/gasoline. Next are the ones that end up as kerosene/diesel. Next are the ones that are marketed as "fuel oil" (but "fuel oil" also generically refers to any petroleum product used for fuel).

(2) These distillates are the main petroleum fuels used in internal combustion engines.

The heavier ones are typically used as fuel for heating, rather than in engines. Diesel engines (typically large engines) running at low RPM and at constant speed can use the heavier distillates (and even light crude).

(3) The remaining part that hasn't boiled off is a "residual oil", and can be used as a fuel (typically for heating).

The viscosity of crude oil and residual oils depends on the composition of the original crude oil. It can have very high viscosity, and be unsuited for use in engines (without pre-heating to reduce viscosity). Sometimes, to make it suitable for use in engines with pre-heating, it is mixed with a distillate to reduce the viscosity.

(4) This mixed fuel - residual fuel oil mixed with distillate fuel oil - is usually used for heating and for some diesel engines.

Essentially, "distillate" is used as a generic term for type (2) above, petroleum distillates. In this sense, gasoline/petrol and diesel fuel are "distillates", and in some countries, "distillate" is a common term for diesel fuel (but not all diesel fuels are petroleum distillates, or even what is usually called "diesel fuel", since biodiesels are not petroleum-derived, and some diesel engines use crude oil or mixed fuels ((4) above)).

In the USA, petroleum distillates heavier than gasoline/petrol (and kerosene?) - i.e., diesel and fuel oil - is classified numerically by viscosity, from 1-6. These are further classified by being specifically intended for use in diesel engines, or heating, or both, and by sulphur content. In this classification, "distillate" refers to the fuels designed for use either in diesel engines or for heating, "diesel" for those intended for use in diesel engines, and "fuel oil" for those intended for heating.

So:

• No. 1 Diesel Fuel: low viscosity fuel intended for use in diesel engines, especially small engines with changing speed and load (e.g., diesel cars).

• No. 1 Fuel Oil: low viscosity fuel oil intended for heating (ideal for portable stoves, outdoor heaters, etc.)

• No. 1 Distillate: low viscosity oil suitable for either of the above two uses.

and so on, for No. 2, etc.

The higher viscosity fuels (i.e., higher numbers) are generally cheaper. As the viscosity increases, they are better suited for larger engines that run at lower RPM and at constant speed and load. Viscosities down to No. 4 still see use in road vehicles (heavy trucks, etc.).

Not only are the higher numbers cheaper, they also tend to burn less cleanly, and produce sootier exhausts. Your father's neighbour was probably using "No. 3 Distillate" or "No. 4 Distillate" (No. 3 disappeared as a category mid-20th century, and was combined with No. 2, but was still used as a category in the 1930s). The stink would be from a high-sulphur version (which would have been cheaper), the stink coming from sulphur compounds in the exhaust (hydrogen sulphide is called "rotten egg gas" for very good (but stinky) reason).

So catalytic cracking didn't really exist until world was 2-ish. Before that when you pumped oil out of the ground you had to distill it to separate it into different grades.

Now keep in mind the goal wasn't originally to get fuel it was to get kerosene for lighting oil. The stuff that was too light was essentially gasoline burned too quickly and not bright enough, it was a waste product. The other waste product was the stuff that was too heavy.

Technically a distillate could be any grade, number 1 grade being gasoline, and number 2 being diesel.

The heavier stuff, 4 through 6 were often called residuals they weren't really used, except for large furnaces, locatives and today large cargo ships. They require preheating to use as they are basically burnable sludge.

Number 3 distillate is probably the candidate, it basically doesn't exist anymore as anything that uses it just uses diesel, it would burn nastier than normal diesel and might be tougher on lighter equipment.

The thing that bugs me is the stories about gasoline being so cheap they would dump it in the rivers to get rid of it. I don't know when that changed. I've seen a bunch on historical oil pricing, but less on the products. So I'm not sure what the cheapest distillate in the 30s would be.