I'm not certain on the L, M & K that you saw listed. But generally we try to carry mostly medium duty tubing onboard for use. For example, a 10mm tube in Light duty rating may have a wall thickness of 1.50 mm, Medium duty would be 2.00 mm, and Heavy being in the 2.5 to 3.00 mm range.
Unless you are pulling hundreds of amps through the lug, I'm sure the differences would never be noticed.
Hell for that matter, I have a lug in service now that I handmade 5 years ago from 19mm tube that continually flows 750 amps and is still alive and kicking.
On paper I'm sure all those letters mean something to some pencil pushing engineer, but in the real world redneck engineering and common sense will still get the job done just fine.
Common wall-thicknesses of copper tubing in the USA are "Type K", "Type L", "Type M", and "Type DWV":
Type K has the thickest wall section of the three types of pressure rated tubing and is commonly used for deep underground burial such as under sidewalks and streets, with a suitable corrosion protection coating or continuous polyethylene sleeve as required by code. In the United States it usually has green colored printing.
Type L has a thinner pipe wall section, and is used in residential and commercial water supply and pressure applications. In the United States it usually has blue colored printing.
Type M has an even thinner pipe wall section, and is used in residential and commercial water supply and pressure applications. In the United States it usually has red colored printing.
Type DWV has the thinnest wall section, and is generally only suitable for unpressurized applications, such as drains, waste and vent (DWV) lines. In the United States it usually has yellow or light orange colored printing, common sizes being 1-¼" , 1-½", and 2" copper tube size.
Types K and L are generally available in both hard drawn "sticks" and in rolls of soft annealed tubing, whereas type M and DWV is usually only available in hard drawn "sticks".