JST:
Oy ! Weh ist Mir ! Sounds quite familiar...
In answer to the OP: "HD" hydraulic oil is likely an ISO 68 oil. This is about a 30 weight oil. Hydraulic oil, in its most basic form, is a mineral based oil with anti foam and corrosion inhibitors. By designation, the basic hydraulic oils fall under an ancient designation of "DTE"- Dynamo, Turbine, Engine. This designation predates the automobile and SAE, let alone ISO ratings for oil viscosity. The old DTE designations for oil 'weights' were: "Light", "Medium", "Heavy-Medium", and "Heavy". Possibly, the OP is looking at a hydraulic oil that would be a DTE "Heavy".
Way Oil has "tackifiers" in it so it will cling to sliding surfaces. The way oil has a high film strength, and is what is needed to maintain an oil film between the carriage saddle and the bedways. On a light duty/small capacity lathe used in a home shop, the ISO 68 hydraulic oil could be used in the carriage and on the sliding surfaces. The "pump" in the carriage that the OP mentions is not intended to lift the carriage saddle to get an oil film established. Usually, the oil pump in the apron of a lathe will dispense the oil into grooves in the carriage 'wings', and the oil then flows onto the bedways and is spread by the movement of the carriage. A thin bodied oil on a lathe doing any kind of heavy work (which is most lathes) will not provide adequate lubrication of the bedways and carriage saddle. In addition, if coolant is used, a light oil will be diluted or washed off the bedways. Way lubes are formulated with tackifiers and heavier body for good reason.
The OP mentions 'fine chips' sticking to the ways if a way lube were to be used. Chips are sizeable, and not likely to be stuck in way lube. If something like lathe filing, or taking a roughing cut on some scaly cast iron or a forging is being done, fine particulate will be coming off the work and onto the lathe bed and cross slide. It is good practice to protect the bedways when this sort of grit or fine chips from mill scale is going to happen. A piece of canvas, cardboard, or whatever is at hand does the trick and catches the fine stuff.
I've taken cuts as fine as 0.001" or a little under (as a spring cut to finish work to dimension). The chips are almost invisible, more like a fine powder deposit on the tool bit. This sort of chip just drops off the toolbit and down between the bedways. It may land on one of the 'girths' (cast bracing between the sides of the lathe bed), or it may land in the chip pan. Get into any kind of a cut, and the chips is going to be heavy enough that it won't 'stick in the way lube'.
Get into some heavier cuts or threading steel on a lathe and the sulphur/lard based cutting oil gets used. Even on a small lathe, cutting oil gets put on with a brush or pump oil can rather than from a circulating coolant system.Get to machining aluminum, and something like kerosene or penetrating oil gets used copiously. If the lathe bed were lubed with hydraulic oil, it would be diluted and thinned to the point of doing little, if any, good, if it were not washed away altogether.
There are many good reasons, well founded in experience over many years, that have resulted in certain types of lubricants being formulated for specific applications. While some old machine tools are so old that no information from the manufacturers exists, there is a wealth of experience and knowledge on the part of the members of this 'board, as well as from lubricant manufacturers. Hydraulic Oils are generally good for plain bearings and gearing running in an oil bath or lubricated by some circulating system. Hydraulic oils see service in tractor trans-axles where they lubricate gearing as well as circulating thru the tractor hydraulic system. However, hydraulic oils just do not have the body or film strength to be used as a way lubricant. I get caught short for way lube, and will use chainsaw 'bar and chain lube' which is about like a light way lube. Bar and Chain lube has the tackifiers and body, and is designed to stay in place in the groove and lands of a chainsaw bar and lube the chain. Sticky, for sure. But, I am not running production work nor are my machine tools in my home shop 'heavy duty".