The rope of a tackle has three named parts
The rope is the fall. One end of it is made fast to one of the blocks, and Steel names the parts from there: “That part of the fall which is fastened to one of the blocks, is called the standing-part, and the other parts of it are called the running-part.” The end the men take hold of is the leading-part, “that part of a tackle which is hauled upon”.
“Fall” was also used loosely for that loose end alone. Steel says so himself: “The rope that connects the blocks of a tackle; but the fall sometimes implies only the loose part which is pulled upon.” Darcy Lever, in The Young Sea Officer’s Sheet Anchor (1808), labels a gun-tackle purchase in just that way: “the standing Part, because it is fixed”, “the running part”, and “the Fall, or part hauled upon”.
In the rest of this tutorial “standing part” is the fixed end of any rope and “leading part” the end that is hauled on, whether or not there is a tackle in between.
Sources. Steel p.176 (tackle, standing and running part), Steel p.168 (leading-part), Steel p.166 (fall); Lever, the page headed “Blocks, Tackles”, about p.15 (the page number cannot be read in the scan).